11th Grade Lesson Plan Packet - Great Hearts Irving ... · 4/11/2020 · ⬜ Sketchbook entry:...
Transcript of 11th Grade Lesson Plan Packet - Great Hearts Irving ... · 4/11/2020 · ⬜ Sketchbook entry:...
11th Grade Lesson Plan
Packet 4/27/2020-5/1/2020
Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming.
April 27-May 1, 2020 Course: 11 Art Teacher(s): Ms. Frank [email protected] Weekly Plan: Monday, April 27 ⬜ Continue working on your project, with attention to craftsmanship in shading, mark-making and line quality. Tuesday, April 28 ⬜ Sketchbook entry: Area of strength and area for improvement ⬜ Continue working on your project Wednesday, April 29 ⬜ Sketchbook entry: Distinctive qualities of your project ⬜ Finish working on project Thursday, April 30 ⬜ Photography Project: Converging Lines and Sense of Place Friday, May 1 ⬜ Complete the formatting of your photography project. If you have time remaining, draw an image that represents your view onto the outside world, whether through a window, from a porch or balcony, or some other vantage point. Use any media you wish. Load a file of your image to the day’s entry. Devote 20 minutes of quality work time to your art assignments each day.
- The assignments will be submitted as a single PDF of photos at the end of the week, with the exception of the photography project, which will be submitted as a Google Slides document.
- For written sketchbook entries you have two options: a. To write them out in your sketchbook as we do in class normally, and include them
in the pdf upload at the end of the week b. To type them into a Google Doc assignment. This will be posted as an “ungraded”
assignment, but really it’s graded as part of the larger packet grade.
Monday, April 27 1. Continue working on your project, with attention to craftsmanship in shading, mark-making and line quality. Keep in mind your individual expressive intentions as you work. As you draw, remember to keep a clean folded piece of paper under your hand to avoid smudging. Tuesday, April 28 1. In a dated sketchbook entry, address an area of strength and an area for improvement in your project. 2. Continue working on your project, using your self-evaluation to help direct your priorities as you work. Wednesday, April 29 1. In a dated sketchbook entry, describe distinctive qualities of your project. What about the drawing style or composition speaks in a particular way of you, your touch, your interests, or your aesthetic judgment? There is a plethora of possibilities, and answering this question involves you looking thoughtfully and receptively at your own work. 2. Complete your project, looking at the image as a whole to achieve unity and harmony, to enhance visual interest, and to accentuate the individual qualities of your work with finishing touches. Thursday, April 30 1. Review the project guidelines for the photographic project, Converging Lines and Sense of Place, and view the demo example on Google Classroom. (Instead of displaying your photographs in a handmade book, you will display them in a Google Slides slideshow presentation.) 2. Proceed to photograph your chosen environment, taking at least 10 quality photographs, from which you will select at least 6 (per guidelines). Friday, May 1 1. Create a Google Slide document for “Converging Lines and Sense of Place” (ready for you on the assignment listing in Google Classroom). Create a title page and a body of 6-8 slides, followed by an end page, into which you could type a message, poem, short story or reflection relating to your project. Save into your GoogleDrive, and upload to the assignment on Google Classroom, then submit.
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Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming.
April 27 - May 1, 2020 Course: 11 Calculus I Teacher(s): Mr. Simmons Weekly Plan: Monday, April 27 ⬜ Read my announcement on Google Classroom’s stream ⬜ Story time! ⬜ Read about suprema ⬜ Complete Problem 1 ⬜ Read the definition of an intersection Tuesday, April 28 ⬜ Read Ch. 11 of Spivak until p. 179 Wednesday, April 29 ⬜ Read Ch. 11, pp. 179-186 Thursday, April 30 ⬜ Read Ch. 11, pp. 186-189 ⬜ Optional proof (ungraded) Friday, May 1 ⬜ Complete Problem 11.6 (“Problem 6”)
Monday, April 27
1. If you have not done so already, please read the long announcement that I posted on the stream (at 8am today).
2. Story time! If it’s technologically feasible to do, please email me at least one sentence letting me know how you’re doing, and tell me a fun story. I miss you (yes, you).
As a way of reviewing and solidifying some of the material from the past few weeks, we are going to be reading Spivak. Some of this is review, so if anyone is bored, I challenge you to prove each of Spivak’s theorems yourself (before looking at Spivak’s proof), and then completing as many as you can of the very fascinating problems he has at the end of each chapter. You’ll enjoy them. In reading Spivak, we are coming in at Ch. 11. He will employ two terms that we have not covered, but which will not take you longer than today’s 40 minutes to become familiar with. In service of preparing us to read Chapter 11, spend today following these instructions:
3. Read about suprema (and infima) on pp. 116-18, starting at the beginning of the chapter and stopping at the statement of Theorem 7-1.
4. On a separate sheet of paper, complete Problem 8.1 (he calls it Problem 1, but to clarify which chapter it’s in, I’m calling it Problem 8.1) at the end of Ch. 8. (You will use this same piece of paper the whole week, turning in just this one item next Sunday.)
5. Read the very brief definition of “intersection” in the middle of p. 43 (in the paragraph that starts, “If f and g are any two functions…”).
Remember to read for understanding. That means possibly pausing on a single sentence and thinking about it for a while, maybe drawing a few diagrams to help you understand it. Don’t rush yourself, or you’ll simply struggle more later. Tuesday, April 28
1. Read Ch. 11, from the beginning of the chapter all the way through the proof of Rolle’s Theorem on p. 179. Read for understanding. (At times, Spivak will write something like, “I leave the remainder of this proof as an exercise for the reader.” Don’t feel obliged to complete those problems (but of course feel free to!).)
Though Spivak doesn’t label it as such, his Theorem 11.2 (he calls it Theorem 2, but to clarify which chapter it’s in, I’m calling it Theorem 11.2) is Fermat’s Theorem. I am aware that he has a slightly different definition of “critical point” from the one we learned. He only mentions points where f prime is zero, whereas we had previously included also points where f prime is undefined (so long as f itself was defined). We will be going with Spivak’s definition, as we will with everything from now on.
Wednesday, April 29
1. Read Ch. 11, until Theorem 11.5 (“Theorem 5”) on p. 186. (At times, Spivak will write something like, “I leave the remainder of this proof as an exercise for the reader.” Don’t feel obliged to complete those problems (but of course feel free to!).)
Thursday, April 30
1. Read Ch. 11, through p. 189. 2. (Optional, ungraded) Page 189 ends with a statement of L’Hopital’s Rule (LOE-pee-Tall’s). If you
would like to, try to prove it. I’m happy to read your proof and give you feedback. (If you’re going to prove it yourself, careful not to glance at spoilers on p. 190.)
Friday, May 1
1. Finish reading Ch. 11 (not including the appendix on convexity and concavity, unless you want to). (At times, Spivak will write something like, “I leave the remainder of this proof as an exercise for the reader.” Don’t feel obliged to complete those problems (but of course feel free to!).)
2. Answer Problem 11.6 (“Problem 6”), writing on Monday’s sheet of paper. 3. If you have extra time of the 40 minutes allotted for math today, spend it mastering the vocabulary
from Ch. 11
Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming.
April 27 - May 1, 2020 Course : 11 Drama Teacher(s) : Mrs. Jimenez ([email protected]) Weekly Plan : Lines should be mastered by the end of this week! Monday, April 27 ⬜ Practice lines for 20 minutes Tuesday, April 28 ⬜ Practice lines for 20 minutes ⬜ 11am-12pm: Zoom Rehearsal - Act 1 (only actors with lines in Act 1 and Student Director) Wednesday, April 29 ⬜ Practice lines for 20 minutes Thursday, April 30 ⬜ Practice lines for 20 minutes Friday, May 1 ⬜ Practice lines for 20 minutes Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________ Student Signature
I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature
Monday, April 27 - Practice lines for 20 minutes. Every day you should review the lines you have already mastered
without looking, then focus on a new scene. If you are a lead and have many lines, choose one scene to review and a new one to work on. Every day you should review a different scene you already have memorized and work for as many days as necessary to master the new one. Record your time on the sheet.
Line memorizing strategies: - Recite your lines OUT LOUD. Practice them like you will say them on stage – projecting,
appropriate speed and emotion, etc. Ask yourself why your character is saying what he/she says and that will help you interpret how to say the line.
- Run your lines with a friend or family member. They should read the lines of the other characters in your scenes while you practice your lines from memory.
- Practice your lines in front of a mirror—the bigger the better! Watch yourself—your facial expressions, how you move, stand, etc.—to be aware of how you look while saying your lines.
- Record yourself saying your lines and listen to the audio (even better if you record your cues!) - Write out your lines by hand (especially if you have a long speech, it is helpful to get it into your
memory through writing it out multiple times). - KNOW YOUR CUES! What line or action comes before you speak? - Run through the parts of the scenes in which you do not speak—what is your character doing
during those parts of the play? - After spending a period of time going over your lines, take a walk or a nap � - REMEMBER: Consistent practice is the key to success!
ALL LINES MUST BE MEMORIZED BY THE END OF THIS WEEK! Tuesday, April 28
- Practice lines for 20 minutes according to Monday’s directions. Record your time on the sheet. - If you are in Act 1 or the Student Director, join the Zoom meeting for online rehearsal from
11am-12pm. This hour will count as your rehearsal time for the week (unless you are woefully behind memorizing lines). Record that you attended the Zoom rehearsal in your sheet to turn in. If you’re not in Act 1, you don’t need to attend; just practice your lines like usual.
- Zoom link: https://zoom.us/j/94257807722?pwd=NDNHWFltclNkZnhCYS9QQzlPWDI3UT09
Wednesday, April 29 - Practice lines for 20 minutes according to Monday’s directions. Record your time on the sheet.
Thursday, April 30 - Practice lines for 20 minutes according to Monday’s directions. Record your time on the sheet.t.
Friday, May 1 - Practice lines for 20 minutes according to Monday’s directions. Record your time on the sheet.
Drama Weekly Line Memorization
Name: Week: 4/27-5/3
Day: Minutes practiced:
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
Minimum time: 20 mins/day, 5 days/week
I verify that this is a true and accurate account of the time I have spent memorizing my lines this past week.
Signature: Date:
Remote Learning Packet
April 27 - May 1, 2020 Course: 11 Greek Teacher(s): Miss Salinas [email protected] Weekly Plan: Monday, April 27 ⬜ If you haven’t already, accept the invitation to join the Google Classroom for 11th Greek! ⬜ Worksheet: Counting; Exercise 8ι ⬜ 40 reps of exercise (20 cardinal, 20 ordinal) Tuesday, April 28 ⬜Worksheet: Exercise 8λ ⬜ 30 reps of exercise (cardinal) Wednesday, April 29 ⬜ Worksheet: 1st half of of ὁ Ὀδυσσευς και ὁ Αἰολος ⬜ 30 reps of exercise (ordinal) Thursday, April 30 ⬜ Worksheet: 2nd half of ὁ Ὀδυσσευς και ὁ Αἰολος ⬜ 40 reps of exercise (cardinal) Friday, May 1 ⬜ Come to Office Hours at 10:30am! (link available in the stream of our Google Classroom) ⬜ Worksheet: Exercise 8θ ⬜ 40 reps of exercise (ordinal)
A note on Google Classroom:
At time of writing this note on Wednesday, 22 April, just over half of the students in this class have
accepted the invitation to join the Google Classroom. Don’t languish in the dark! If you are having
trouble accessing it, please reach out to me or Ms. Weatherton, our registrar, who can help get you set up.
This week (Monday, April 27, when you are reading this), our work is completely posted on Google
Classroom in handy-dandy daily assignments you can complete without ever having to print and scan
anything, along with links to Friday’s Office Hours. If you are one of the nine students who has not
done so, please log in to your Google Classroom as soon as you have access to a computer and accept
the invitation to join 11th Greek!
A note on the second exercise (heh, literally) of each day:
This week, one of my favorite youtube channels that makes content related to teaching classical languages
(I’m aware of my critical level of nerdery) posted a video with tips on how to master counting in foreign
languages. What luck! His pro-tip is this: if you use the numbers of your target language to count reps of
exercise, your brain is much more likely to latch on to them. (Pending permission, I’ll link the video in
our Google Classroom. He’s better at explaining things than I am.) Our second assignment of each day
this week is this: do a number of reps of your choice of the following: push-ups, sit-ups, squats,
step-ups, calf raises, crunches, plans (in seconds), or wall-sits (in seconds) and count them in Greek.
If you need to make a paper copy of the numbers to hold or put in front of you for the first few days, do!
A note on sanity levels:
How are you? Are you staying sane? Are you either bored and unchallenged or overwhelmed? Are the
days of the week losing all meaning? Is time losing all meaning? Is meaning losing all meaning? Same. I
care about your sanity, especially as Greek relates to it. If you would like more of a challenge or if the
workload is getting to be too much, reach out and let me know. I’m creating these lessons week by
week, so I have the flexibility to adapt subsequent plans according to how the class is doing. So far it
seems alright, but if you have feedback, shoot it to me. Hey, you know a good way to send me feedback?
Logging on to your computer and emailing me! While you’re there, accept the invitation to join the
Google Classroom! (Okay, I’ll stop...I realize I may be a cause of your potentially low sanity levels.)
Monday, April 27
➔ Complete Monday’s one-page worksheet, which includes counting and Exercise 8ι.
➔ Today, do forty reps of your exercise of choice, counting twice to δεκα (cardinal numbers) and twice to δεκατος (ordinal numbers).
Tuesday, April 28
➔ Complete Tuesday’s one-page worksheet, which includes Exercise 8λ. Τhis worksheet is best done on Google Classroom.
➔ Today, do thirty reps of your exercise of choice, counting thrice to δεκα (cardinal numbers).
Wednesday, April 29
➔ Complete Wednesday’s two-page worksheet, which includes the first half of a story about Odysseus. Τhis worksheet is best done on Google Classroom.
➔ Today, do thirty reps of your exercise of choice, counting thrice to δεκατος (ordinal numbers).
Thursday, April 30
➔ Complete Thursday’s two-page worksheet, which includes the second half of a story about Odysseus. Τhis worksheet is best done on Google Classroom.
➔ Today, do forty reps of your exercise of choice, counting four times to δεκα (cardinal numbers).
Friday, May 1
➔ Come to Office Hours at 10:30am! Link available in the stream of our Google Classroom. Ask questions, give me feedback on your workload and the best format for worksheets, or at least connect on a human level with someone you’re not quarantined with.
➔ Complete Friday’s two-page worksheet, which includes Exercise 8θ. Τhis worksheet is best done on Google Classroom.
➔ Today, do forty reps of your exercise of choice, counting four times to δεκατος (ordinal numbers).
Monday
Counting to δεκα / Exercise 8ι Write the cardinal and ordinal numbers in Greek and English here.
CARDINAL
1.
2.
3.
4. τετταρες (m./f.), τετταρα (n.) - four
5.
6.
7.
8.
9.
10.
ORDINAL
1.
2.
3.
4. τεταρος, τεταρη, τεταρον - fourth
5.
6.
7.
8.
9.
10.
Exercise 8ι (from workbook)
Select the English equivalent from the word bank above and write it on the line in front of each Greek numerical adjective.
1. _______________ τεταρτος
2. _______________ ἑπτα
3. _______________ ἐνατος
4. _______________ ἑξ
5. _______________ δεκατος
6. _______________ ὀκτω
7. _______________ δευτερος
8. _______________ τρεις
9. _______________ δεκα
10. _______________ πεμπτος
11. _______________ ὀγδοος
12. _______________ πεντε
13. _______________ ἑν
14. _______________ τριτος
15. _______________ πρωτος
16. _______________ εἰς
17. _______________ δυο
18. _______________ ἑβδομος
19. _______________ ἑκτος
20. _______________ τετταρες
21. _______________ ἐννεα
22. _______________ μια
Tuesday
Exercise 8λ Τhis worksheet is best done on Google Classroom. On the space to the right of each sentence, translate the underlined phrase. Then, circle the type of time expression used.
1. We worked τρεις ἡμερας. ______________________
gen. of time within which dat. of time when acc. duration of time
2. It all took place τριῶν ἡμερῶν . ______________________
gen. of time within which dat. of time when acc. duration of time
3. She arrived τῇ τριτῇ ἡμερᾳ . ______________________
gen. of time within which dat. of time when acc. duration of time
4. They had the work finished πεντε ἡμερῶν . ______________________
gen. of time within which dat. of time when acc. duration of time
5. πασαν την ἡμεραν the farmer was working. ______________________
gen. of time within which dat. of time when acc. duration of time
6. The athlete received a prize τῇ πεμπῃ ἡμερᾳ . ______________________
gen. of time within which dat. of time when acc. duration of time
7. τῇ ὑστεραιᾳ they left for the harbor. ______________________
gen. of time within which dat. of time when acc. duration of time
8. They traveled πολυν χρονον. ______________________
gen. of time within which dat. of time when acc. duration of time
Wednesday
ὁ Ὀδυσσευς και ὁ Aἰολος - ἡμισυ πρωτον (“1st half”) This worksheet is best done on Google Classroom. Read the following passage and answer the comprehension questions.
Odysseus tells how he sailed on to the island of Aeolus, king of the winds, and almost reached home:
ἐπει δε ἐκ του ἀντρου ( the cave) του Κυκλωπος ἐκφευγομεν, ἐπανερχομεθα ( we return)
ταχεως προς τους ἑταιρους. οἱ δε, ἐπει ἡμας ὁρωσιν, χαιρουσιν. τῇ δ’ὑστεραιᾷ κελευω
αὐτους εἰς την ναυν αὐθις εἰσβαινειν. οὑτως οὐν ἀποπλεομεν.
1. What do Odysseus and his men do when they escape from the cave of the Cyclops?
_____________________________________________________________________________________
_____________________________________________________________________________________
2. What does Odysseus order his men to do the next day?
_____________________________________________________________________________________
_____________________________________________________________________________________
3. τῇ δ’ὑστεραιᾷ is which of the following?
gen. of time within which dat. of time when acc. duration of time
δι’ὀλιγου δε εἰς νησον Αἰολιαν ( Aeolia - the island of Aeolus, king of the winds)
ἀφικνουμεθα. ἐκει δε οἰκει ὁ Αἰολος, βασιλευς των ἀνεμων ( of the winds). ἡμας δε
εὐμενως ( kindly) δεχομενος πολυν χρονον ξενιζει (entertains). ἐπει δε ἐγω κελευω αὐτον
ἡμας ἀποπεμπειν, παρεχει μοι ἀσκον ( bag) τινα, εἰς ὅν ( which) παντας τους ἀνεμους
καταδει (he ties up) πλην (+gen, except) ἑνος, Ζεφυρου πραου ( gentle).
4. Where do Odysseus and his men arrive next?
_____________________________________________________________________________________
_____________________________________________________________________________________
5. How long do Odysseus and his men stay with Aeolus?
_____________________________________________________________________________________
_____________________________________________________________________________________
6. What does Aeolus give Odysseus at his departure?
_____________________________________________________________________________________
_____________________________________________________________________________________
7. What wind was not in the bag?
_____________________________________________________________________________________
_____________________________________________________________________________________
Thursday
ὁ Ὀδυσσευς και ὁ Aἰολος - ἡμισυ δευτερον (“2nd half”) This worksheet is best done on Google Classroom. Read the following passage and answer the comprehension questions.
ἐννεα μεν οὐν ἡμερας πλεομεν, τῇ δε δεκατῇ ὁρωμεν την πατριδα γην ( our fatherland).
ἐνταυθα δη ἐγω καθευδω. οἱ δε ἑταιροι, ἐπει ὁρωσι με καθευδοντα ( sleeping), οὑτω
λεγουσιν. “τι ἐν τῷ ἀσκῷ ( bag) ἐνεστιν; πολυς δηπου ( surely) χρυσος (gold) ἐνεστιν, πολυ
τε ἀργυριον ( silver), δωρα (gifts) του Αἰολου. ἀγετε δη ( come on!), λυετε τον ἀσκον και
τον χρυσον αἱρειτε ( take).”
1. How long do Odysseus and his men sail for?
_____________________________________________________________________________________
2. ἐννεα...ἡμερας is which of the following?
gen. of time within which dat. of time when acc. duration of time
3. τῇ δε δεκατῇ is which of the following?
gen. of time within which dat. of time when acc. duration of time
4. When they come within sight of their fatherland, what does Odysseus do?
_____________________________________________________________________________________
_____________________________________________________________________________________
5. What do his comrades think is in the bag?
_____________________________________________________________________________________
_____________________________________________________________________________________
ἐπει δε λυουσι τον ἀσκον, εὐθυς ( at once) ἐκπετονται ( fly out) παντες οἱ ἀνεμοι και
χειμωνα δεινον ποιουσι και την ναυν ἀπο της πατριδος γης ἀπελαυνουσιν ( they expel/push
away). ἐγω δε ἐγειρομαι ( wake myself up) και γιγνώσκω τι γιγνεται (happens). ἀθυμω ( I
despair) οὐν και βουλομαι ῥιπτειν ( to throw) ἐμαυτον ( myself) εἰς την θαλατταν. οἱ δε
ἑταιροι σῳζουσι με. οὑτως οὐν οἱ ἀνεμοι ἡμας εἰς την του Αἰολου νησον παλιν ( again)
φερουσιν.
6. What happens when the men open the bag?
_____________________________________________________________________________________
_____________________________________________________________________________________
7. How does Odysseus react when he wakes up?
_____________________________________________________________________________________
_____________________________________________________________________________________
8. Why doesn’t Odysseus carry out his plan of despair?
_____________________________________________________________________________________
_____________________________________________________________________________________
9. Where do the winds carry the ship at last?
_____________________________________________________________________________________
_____________________________________________________________________________________
Friday
Exercise 8θ
This worksheet is best done on Google Classroom. Answer the questions and translate the following sentences, which continue the story from Odysseus’ perspective.
1. ἐπει εἰς την νησον ἀφικνουμεθα, προς τον του Αἰολου οἰκον ἐρχομαι.
a. Which is correct about ἀφικνουμεθα?
2nd person sg: you arrive 1st person pl: we arrive 2nd person pl: y’all arrive
b. What case is του Αἰολου, and what keyword thus goes before it when translating?
gen: of dat: in, on acc: no keyword
Translation: _____________________________________________________________________
2. ὁ δε, ἐπει ὁρᾷ με, μαλα θαυμαζει και, “τι πασχεις;” φησιν, “τι αὐθις παρει;”
a. Are the verbs ὁρᾷ, θαυμαζει, and πασχεις in the active or middle voice?
active middle
b. Which punctuation mark is ; in Greek equivalent to in English?
- : ? ! , .
Translation: _____________________________________________________________________
3. ἐγω δε ἀποκρινομαι, “οἱ ἑταιροι αἰτιοι ( to blame) εἰσιν. τους γαρ ἀνεμους ( the
winds) ἐλυσαν ( they loosed). ἀλλα βοηθει ἑμιν, ὠ φιλε.”
a. Which of the following does NOT describe Odysseus’ tone?
suppliant/begging responsible/mature blaming/selfish
b. What little thing does Odysseus do in the last sentence to appeal to Aeolus and attempt to make him more willing to help? _____________________________________________________________________
Translation: _____________________________________________________________________
4. ὁ δε Αἰολος, “ἀπιτε (go away) ταχεως,” φησιν, “ἀπο της νησου. οὐ γαρ δυνατον
ἐστιν ὑμιν βοηθειν. οἱ γαρ θεοι δηπου (surely) μισουσιν (hate) ὑμας.”
a. Does Aeolus want to help Odysseus or not?
of course, yes! not at all
b. What mood is ἀπιτε?
indicative imperative infinitive subjunctive
Translation: _____________________________________________________________________
Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming.
April 27 - May 1, 2020 Course: 11 Humane Letters Teacher(s): Mr. Brandolini [email protected]
Mr. Mercer [email protected] Welcome to Week 5! In the hopes of providing the greatest opportunity to read Hamlet with the leisure and appreciation it deserves, we will be scaling back the written work of this week to primarily focus on enjoying this timeless cornerstone of the Western Tradition. Daily comments for each reading are included, kept deliberately brief to serve as signposting for major themes/moments to keep an eye out for before taking on the day’s reading. For this week, there will only be two reading quizzes, on Wednesday and Friday. We will revisit Hamlet for a deeper essay at the start of next week. Weekly Plan: Monday, April 27 ⬜ Read William Shakespeare, Hamlet, Act 1 (p. 3-69 of the Folger edition) Tuesday, April 28 ⬜ Read Hamlet, Act 2 (p. 73-119) Wednesday, April 29 ⬜ Complete Quiz on Hamlet, Acts 1&2 ⬜ Read Hamlet, Act 3 (p. 123-185) Thursday, April 30 ⬜ Read Hamlet, Act 4 (p. 189-235) Friday, May 1 ⬜ Read Hamlet, Act 5 (p. 239-287) ⬜ Complete Quiz on Hamlet, Acts 3-5 Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________Student Signature
I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature
Monday, April 27 Read William Shakespeare, Hamlet, Act 1 (p. 3-69 of the Folger edition)
At long last, we come to the end of our series of tragedies! It is fitting that we should end with the greatest of all, Shakespeare’s Hamlet. If Sophocles’ Oedipus exemplifies the perfect distillation of Aristotle’s standards for tragic action, then Hamlet reinvigorates and elevates the movement to heights that have yet to be surpassed. In the most general sense, a tragic tale has three distinct parts:
1) a Fall that communicates the primary flaw, motive, or concern of the play; 2) a period of great stasis and suffering, or a Wandering; 3) a final Reconciliation, or a resolution of the play’s action that leaves the hero or the cosmos of the story in a greater order than it started. Thus far, we have seen that each play can tackle these stages to varying degrees of emphasis; in
Oresteia, each play essentially serves as an emphasis on each respective stage, while also containing a complete movement within each individual work. In Oedipus, the vast majority of emphasis is in the first aspect, the Fall of the king and its causes. Hamlet is somewhat unusual in that it places primary emphasis on the latter two, leaving much of the action of the Fall to before the start of the drama. A key reason for this is that Hamlet is written as much in a Christian tradition as it is in a literary one; by this I mean that just as Homer, Aeschylus, and Sophocles innately concerned with the traditions and cosmology of their society, so too is Hamlet directly informed by a Christian world: it will be especially important to read the play with an eye towards the redemptive aspect of the play.
For the time being, however, simply begin! Avoid the temptation to over-analyze or to write off the sufferings and actions of the characters as merely confused or selfish; either extreme will distract you from simply opening yourself to the play’s splendor. Tuesday, April 28 Read Hamlet, Act 2 (p. 73-119)
“Something is rotten in the state of Denmark” says Marcellus in Act 1.4. As we continue onward into Act 2, we should hopefully begin to see that this is very much apparent. One of the prevailing themes of the play is infection, sickness of soul, and sin, the “vicious mole of nature” in men since birth (1.4.26); according to the Ghost in 1.5, the very cause of the descent of Denmark is due to his death, wherein “sleeping in my orchard, / A serpent stung me. So the whole ear of Denmark / is by a forged process of my death / rankly abused.” (1.5.42-45). In this sense, the “Fall” of the drama is the Fall of Original Sin: Hamlet is plagued by the problem of sin and guilt, particularly as he is given the apparently impossible task of getting revenge for his father and remaining pure (much like Orestes’ mandate from Apollo). Pay attention to imagery of infection and gardens.
As we continue into Act 2, remember Hamlet’s plan to put on an “antic disposition”: pay special attention to his different moods and behaviors, namely when performing before others, when confiding in a single person, or when alone and engaging in a soliloquy. One of the play’s greatest ambiguities is to
what extent Hamlet’s antic disposition is an act. Hamlet very often discusses or emphasizes knowledge and ignorance: why is he so concerned with these, and do they indicate his tragic hamartia in any way? Wednesday, April 29 Complete Quiz on Hamlet, Acts 1&2 Read Hamlet, Act 3 (p. 123-185)
Hamlet’s famous reflection on the nature of suffering: what is it that paralyzes him so? Act 2 revealed that he has waited two months since the Ghost’s command. Notice that Hamlet does not ask whether “To live”, but rather “To be": his concern is existential, pondering whether man ought to exist at all, and is not merely born out of personal depression. His conversation with Ophelia following this soliloquy is also revealing of his assessment of human nature. Why does he speak to Ophelia the way he does? Is he trying to tell her something? How could these opening scenes be related to his concern over knowledge vs. ignorance?
Pay attention to how Hamlet reacts when approaching Claudius: we see both at their lowest and most desperate here, albeit in a somewhat ironic situation. As you continue reading, remember this moment and consider the consequences of Hamlet’s decision. His interaction with his mother, meanwhile, provides a stark contrast with our previous tragic heroes: Hamlet refuses to cross any “natural” boundaries when dealing with his mother, and instead seeks to help her by revealing the severity of her sin and the state of her soul to herself. Perhaps some good will come of Hamlet’s antic disposition? Thursday, April 30 Read Hamlet, Act 4 (p. 189-235)
Hamlet’s disposition has clearly changed--what has caused this? Does he feel any remorse or
regret for what he has done in Act 3? Has he become no better than Claudius himself? The state Hamlet leaves Denmark in under Claudius as he leaves for England is dire; Claudius will put the blame on Hamlet, but is this not also an indication of Claudius’ own inability to rule well? Notice also the stark contrast in the ways Laertes and Ophelia react to their father’s death. To what extent does Laertes’ thirst for revenge serve as a foil to Hamlet’s? Friday, May 1 Read Hamlet, Act 5 (p. 239-287) Complete Quiz on Hamlet, Acts 3-5
By the end, has Hamlet had any reconciliation, any change of heart? Like much of the play, it is ambiguous, but the most critical passage begins at line 5.2.233, where we see Hamlet answer the question posed earlier of whether or not “To be”. Is this moment a despairing admission of defeat, or a surrendering himself to God’s Providence? Think back to the theme of Denmark and Original Sin: has
Hamlet managed to reconcile his understanding of the problem of sin? The play leaves Denmark’s fate uncertain: has the rot been purged? Has Hamlet saved his family and his kingdom by revealing their errors? Once you’ve read, take some time to see if you can answer some of these questions for yourself. Next week, we will begin with a short essay that will attempt to assess these developments and see how they synthesize with the traditional, Aristotelian understanding of the tragic action.
__________________________________ __________________________________ Wednesday April 29 __________________________________ Reading Quiz: Hamlet, Acts 1&2 __________________________________ 1. What or who is it that Barnardo, Marcellus, and Horatio encounter during the night watch in scene 1?
A. a wild beast B. a royal ghost C. a celestial sign D. a fleeing criminal
2. Which of the three characters is initially doubtful of their encounter?
A. Horatio B. Barnardo C. Marcellus
3. What proposal does Horatio make at the end of scene 1?
A. to make known to Hamlet what they encountered in the night B. to make known to Hamlet Ophelia’s love for him C. to make known to Hamlet his uncle’s misdeeds D. to make known to Hamlet a plot against his life
4. Identify the speaker: “Do not forever with thy vailed lids / Seek for thy noble father in the dust. / Thou know’st ‘tis common; all that lives must die, / Passing through nature to eternity.”
A. King Claudius B. Polonius C. Hamlet D. Queen Gertrude
5. From where do Claudius and Gertrude try to persuade Hamlet from going?
A. London B. Norway C. Wittenburg D. Paris
6. With regard to what does Laertes tell Ophelia to exercise utmost caution?
A. Hamlet’s love toward her B. Claudius’s love toward her C. her father’s counsel D. her own judgement
7. Identify the speaker: “This above all: to thine own self be true, / And it must follow, as the night the day, / Thou canst not then be false to any man.”
A. King Claudius B. Laertes C. Polonius D. Horatio
8. What does the spirit of young Hamlet’s father reveal to him in the night?
A. that the kingdom of Denmark will be overrun by Norsemen B. that Polonius was plotting to have Laertes become the next king C. that Gertrude was not his real mother D. that Claudius murdered him to take the throne and the queen
9. What does Hamlet demand of Horatio and Marcellus at the end of the first act?
A. their allegiance to him in battle B. a vow of silence about what they saw that night C. money for him to travel to England D. their help solving a riddle
10. What does Ophelia report to her father in the opening scene of the second act?
A. Hamlet’s strange behavior toward her B. suspicious activity among Hamlet’s friends C. that she’s being courted by an English prince D. her ambition to become queen of Denmark
11. What does Claudius call on Rosencrantz and Guildenstern to do?
A. discover the source of treachery in the palace B. find out why Hamlet has changed and try to cheer him up C. assassinate the prince of Norway D. spy on Laertes in France
12. Identify the speaker: “O, what a rogue and peasant slave am I!”
A. Polonius B. Rosencrantz C. Guildenstern D. Hamlet
13. In two to three complete sentences, answer the following question: Does Hamlet have an optimistic or pessimistic view of the human condition? _____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
__________________________________ __________________________________ Friday May 1 __________________________________ Reading Quiz: Hamlet, Acts 3-5 __________________________________ 1. Identify the speaker: “And for your part, Ophelia, I do wish / That your good beauties be the happy cause / Of Hamlet’s wildness. So shall I hope your virtues / Will bring him to his wonted way again, / To both your honors.”
A. Polonius B. Queen Gertrude C. Laertes D. King Claudius
2. What does Hamlet make known to Ophelia in the first scene of Act 3?
A. that he was the cause of his father’s death B. that she was the cause of his father’s death C. that he once hated her but now loves her D. that he once loved her but no longer does
3. How does Claudius respond to what he sees of the play The Murder of Gonzago ?
A. he expresses remorse over his grave offense B. he shows pride in the fact that he was not caught in his misdeeds C. he gives a prize to the players for their skillful acting D. he banishes the players from the realm for their poor acting
4. Who does Hamlet strike dead before knowing his identity?
A. Claudius B. Polonius C. Laertes D. one of the players
5. Identify the speaker: “A man may fish with the worm that hath eat of a king and eat of the fish that hath fed of that worm.”
A. Polonius B. Claudius C. Rosencrantz D. Hamlet
6. To where does Claudius seek to send Hamlet?
A. England B. Norway C. Poland D. France
7. Whom does the “rabble” threaten to make king? A. Hamlet B. Horatio C. Laertes D. Polonius
8. Who captures Hamlet and sends him back to Denmark?
A. Fortinbras B. an unknown pirate C. English soldiers D. Polacks
9. What plan is hatched to ensure Hamlet is killed by “accident”?
A. put him to sea in a ship that isn’t seaworthy B. poison him in his sleep C. place a candle in an unsafe place near his quarters D. convince him to agree to a friendly bout of fencing with Laertes
10. Whose grave is being prepared at the beginning of Act 5?
A. Ophelia’s B. Polonius’s C. Hamlet’s D. Horatio’s
11. Identify the speaker: “I am justly killed with mine own treachery.”
A. Claudius B. Osric C. Hamlet D. Laertes
12. True or false: Hamlet and Laertes make amends in their final moments.
A. True B. False
13. In two to three sentences, answer the following question: What exactly is Hamlet asking when he poses the question, “To be or not to be” (3.1.64)? _____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
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Remote Learning Packet
April 27-May 1, 2020 Course: 11 Precalculus Teacher(s): Mr. Simmons Weekly Plan: Monday, May 4 ⬜ Story time! ⬜ Read “Right triangles” ⬜ Problems 1-5 odd Tuesday, May 5 ⬜ Read “The Sine, Cosine, and Tangent functions” Wednesday, May 6 ⬜ Problems 1-5 (all) Thursday, May 7 ⬜ Read “Relationships between Trig Functions” Friday, May 8 ⬜ Problems 1-8 (all)
Monday, May 4
1. Story time! If technologically feasible, send me an email telling me a fun story that happened recently. I miss you! (yes, you)
Learning from packets is much harder than learning in class. I’ve found (after much searching) a good resource for learning trigonometry. We’ll be using it for the remainder of the year. It’s the second part of a book called Precalculus by Mr. Joseph Gerth. I’ve included it as a material on Google Classroom.
2. Read “Right triangles” from Gerth’s Precalculus (pp. 84-90). 3. Complete problems 1, 3, and 5 (pp. 90-91)
Tuesday, May 5
1. Read “The Sine, Cosine, and Tangent functions” (pp. 93-109). Wednesday, May 6
1. Complete problems 1-5 (pp. 109-110) on the same piece of paper from Monday. Thursday, May 7
1. Read “Relationships between the Trig functions” (pp. 113-121) Friday, May 8
1. Complete problems 1-8 (pp. 121-122) on the same piece of paper from Monday and Wednesday.
82
Part
two:
Tri
gono
met
ry
Trig
onom
etry
is th
e st
udy
of a
ngle
s an
d th
eir r
elat
ions
hip
to tr
iang
les.
As y
ou’ll
soo
n se
e,
we
atte
mpt
to
rela
te t
he a
ngle
s of
a t
riang
le t
o its
sid
es. I
t is
per
haps
uns
urpr
ising
tha
t th
is Pa
rt is
clo
sely
rel
ated
to
Geo
met
ry –
we’
ll as
k th
at f
amou
s qu
estio
n, “
Wha
t is
the
rela
tions
hip?
” W
hat
is in
tere
stin
g, h
owev
er,
is th
at t
his
Part
is
also
clo
sely
rel
ated
to
Alge
bra.
We
will
ofte
n w
ork
with
equ
atio
ns a
nd u
se a
lgeb
ra t
o ch
ange
the
m i
nto
som
ethi
ng d
iffer
ent.
This
reve
als o
ne re
ason
why
Trig
onom
etry
is st
udie
d: I
t is,
in so
me
sens
e, a
uni
on o
f tho
se
two
grea
t dis
cipl
ines
of G
eom
etry
and
Alg
ebra
. It t
akes
wha
t you
kno
w in
eac
h di
scip
line
and
appl
ies
it, s
o th
at y
ou c
an e
xpan
d ea
ch.
This
bran
ch o
f m
athe
mat
ics
is a
lso
stud
ied
at t
his
poin
t be
caus
e it
is a
ver
y im
port
ant
Calc
ulus
con
cept
. Cal
culu
s tea
cher
s and
text
book
s will
ass
ume
that
you
can
eva
luat
e, e
.g.,
sin𝜋 4 in
sec
onds
. You
will
also
be
expe
cted
to
have
dee
p in
sight
into
the
gra
phs
of t
he
Trig
onom
etric
func
tions
. Sin
ce y
ou’ll
ofte
n be
find
ing
the
area
of a
Trig
onom
etric
func
tion
or, p
erha
ps, d
eter
min
ing
its b
ehav
ior a
s it
goes
to in
finity
.
Man
y st
uden
ts s
trug
gle
with
Trig
onom
etry
. Thi
s is
mos
tly li
kely
due
to th
eir d
efic
ienc
ies
in G
eom
etry
, Alg
ebra
, or
both
. We
will
giv
e a
smal
l rev
iew
of
right
tria
ngle
Geo
met
ry,
whi
ch w
e th
ink
will
be
help
ful f
or y
ou. B
ut m
any
of y
ou m
ay n
eed
to d
o so
me
read
ing,
re
sear
ch, a
nd p
ract
ice
on y
our o
wn
to g
et y
ou u
p to
par
. Eve
n w
hen
you
are
wel
l-ve
rsed
in
the
pre-
requ
isite
s, yo
u m
ust a
lso b
e pr
epar
ed to
spe
nd ti
me
prac
ticin
g an
d m
aste
ring
new
tech
niqu
es a
s w
ell.
Anot
her
issue
tha
t yo
u m
ust
be a
war
e of
is t
hat
Trig
onom
etry
is v
ery
vert
ical
. Tha
t is
, an
ythi
ng le
arne
d se
rves
as
a fo
unda
tion
for t
he n
ext t
hing
. If y
ou d
on’t
lear
n so
met
hing
, th
eref
ore,
it’s
near
ly im
poss
ible
to p
rogr
ess.
The
corn
erst
one
of th
is Pa
rt is
mos
t ass
ured
ly
right
tria
ngle
s. Yo
u sim
ply
mus
t m
aste
r th
e sp
ecia
l rig
ht t
riang
les
pres
ente
d in
the
firs
t se
ctio
n. F
ailu
re to
do
so w
ill n
early
gua
rant
ee fa
ilure
. As
such
, you
mus
t app
roac
h ea
ch
sect
ion
with
due
dili
genc
e, a
nd m
aste
r any
and
all
of th
e co
nten
t.
§1
Lin
ear f
unct
ions
– 8
3
Uni
t fou
r Ri
ght t
riang
le T
rigon
omet
ry
Τριγωνομετρία
– G
reek
for
Tri
gon
omet
ry. L
iter
ally
mea
nin
g “t
rian
gle
mea
suri
ng.
”
Uni
t fou
r – 8
4
Trig
onom
etry
is a
bra
nch
of m
athe
mat
ics
that
look
s at
tria
ngle
s, an
gles
, and
circ
les.
It’s
a co
mpl
ex a
nd o
ften
rigor
ous
art
that
req
uire
s in
tuiti
on o
ver
skill
s. It
is i
mpe
rativ
e to
de
velo
p an
d m
aste
r th
e fo
unda
tiona
l kno
wle
dge
of T
rigon
omet
ry, b
ecau
se w
ithou
t it,
su
cces
s is
ver
y di
fficu
lt.
Alth
ough
we
assu
me
the
read
er k
now
s no
thin
g of
Trig
onom
etry
, it m
ight
be
wise
to h
ave
som
e ex
perie
nce
befo
re b
egin
ning
thi
s U
nit.
Addi
tiona
lly,
ther
e ar
e a
few
Geo
met
ry
theo
rem
s tha
t are
ass
umed
to b
e kn
own.
If a
ny o
f the
follo
win
g se
ems d
iffic
ult,
it is
high
ly
reco
mm
ende
d to
con
sult
the
prev
ious
text
s.
We
begi
n by
stu
dyin
g so
me
basic
Geo
met
ry.
Afte
r th
is, w
e th
en i
ntro
duce
the
bas
ic
Trig
onom
etric
func
tions
: Si
ne, c
osin
e, a
nd ta
ngen
t.
§1
Ri
ght t
riang
les
Sinc
e th
e w
ord
“Trig
onom
etry
” lite
rally
mea
ns “t
riang
le m
easu
re”,
perh
aps i
t isn
o su
rpris
e th
at w
e be
gin
wor
king
with
tria
ngle
s. Re
call
that
a tr
iang
le is
a p
olyg
on w
ith th
ree
side
s, an
d th
at a
rig
ht t
riang
le is
so
calle
d be
caus
e on
e of
its
angl
es m
easu
res
90
° (w
hich
we
defin
e as
a r
ight
ang
le).
Thus
, Fig
ure
21 is
a ri
ght t
riang
le.
This
simpl
e ob
ject
con
tain
s m
any
prof
ound
trut
hs, s
ome
of w
hich
you
may
hav
e le
arne
d in
Geo
met
ry c
lass
. Pe
rhap
s th
e m
ost
wel
l-kno
wn
and
prof
ound
tru
th i
s kn
own
as
Pyth
agor
as’ T
heor
em. I
t has
bee
n kn
own
and
used
for m
illen
nia,
con
tain
s m
ore
than
30
0
diffe
rent
pro
ofs,
and
is us
ed in
just
abo
ut e
very
bra
nch
of m
athe
mat
ics.
i
Pyth
agor
as’ T
heor
em
Giv
en a
rig
ht t
riang
le w
ith le
ngth
s 𝑎
,𝑏,𝑐
, whe
re 𝑐
is t
he h
ypot
enus
e an
d 𝑎
,𝑏 a
re t
he
legs
, the
n 𝑎
2+𝑏
2=𝑐
2.
i E
ven
stat
istic
s ha
s a
use
for P
ytha
gora
s’ Th
eore
m! Figu
re 2
1
All r
ight
ang
les
will
hav
e a
box
(like
the
one
next
to 𝐵
) to
deno
te th
e fa
ct th
at th
ey a
re 9
0°.
Reca
ll th
at it
is n
ot g
ood
enou
gh to
“loo
k lik
e” it
’s 9
0° -
itm
ust h
ave
the
box.
§1
Rig
ht tr
iang
les
– 85
We
will
usu
ally
pro
ve e
ach
basi
c an
d fo
unda
tiona
l ass
ertio
n in
this
cour
se. H
owev
er, w
e be
lieve
that
mos
t will
hav
e al
read
y pr
oved
this
theo
rem
on
thei
r ow
n. T
here
fore
we
will
no
t pro
ve th
is as
sert
ion
here
.ii
Exam
ple
1
Find
the
leng
th o
f the
miss
ing
side
in F
igur
e 22
.
The
hypo
tenu
se –
alw
ays
oppo
site
of t
he r
ight
ang
le –
is s
ide
leng
th 𝐴𝐵
, whi
ch h
as a
le
ngth
of 3
1. T
hus
we
can
say
that
𝑐, w
hich
rep
rese
nts
the
leng
th o
f the
hyp
oten
use
in
Pyth
agor
as’ T
heor
em, i
s 3
1. A
nd s
ince
𝐴𝐶
is o
ne o
f our
legs
, we
will
let
𝑎=
11
.
Giv
enth
is, w
e no
w h
ave
11
2+𝑏
2=
31
2
12
1+𝑏
2=
96
1.
This
is n
ow a
sim
ple
alge
bra
prob
lem
. Sol
ving
for 𝑏
we
get
𝑏2
=84
0
𝑏=2
8.9
8.
Som
e rig
ht tr
iang
les
are
spec
ial;
they
com
e in
kno
wab
le ra
tios.
And
if w
e kn
ow th
e ra
tios
of a
ll th
ree
sides
, the
n w
e do
not
nee
d to
use
Pyt
hago
ras’
Theo
rem
at a
ll.
Cons
ider
the
right
tria
ngle
sho
wn
in F
igur
e 23
.
ii T
he in
tern
et is
a g
reat
reso
urce
. Use
it.
Figu
re 2
2
Pict
ures
are
not
(and
nev
er w
ill b
e) d
raw
n pe
rfect
ly to
sca
le.
Uni
t fou
r – 8
6
This
is a
n is
osce
les
tria
ngle
, whi
ch m
eans
tha
t tw
o of
the
sid
e le
ngth
s ar
e co
ngru
ent.
Cons
eque
ntly
, we
can
mak
e th
e fo
llow
ing
asse
rtio
n.
Isos
cele
s rig
ht tr
iang
le s
ide
ratio
G
iven
an
isosc
eles
righ
t tria
ngle
with
a le
g of
leng
th𝑠,
then
the
othe
r leg
has
a le
ngth
of
𝑠 a
nd th
e hy
pote
nuse
has
a le
ngth
of 𝑠√2.
Proo
f.
Cons
truc
t iso
scel
es ri
ght t
riang
le 𝐴𝐵𝐶
with
leg 𝐴𝐵
=𝑠.
The
n 𝐴𝐶
=𝑠
beca
use
we
have
an
isosc
eles
righ
t tria
ngle
.
Furt
her,
we
know
that
the
leng
th o
f the
hyp
oten
use
is
𝐴𝐵
2+𝐴𝐶
2=𝐵𝐶
2
by P
ytha
gora
s’ Th
eore
m.
But
we
know
the
len
gth
of b
oth 𝐴𝐵
and
𝐴𝐶
, an
d so
, by
su
bstit
utio
n, w
e ha
ve
𝑠2
+𝑠
2=𝐵𝐶
2.
Whe
nce,
afte
r sim
plifi
catio
n, w
e ha
ve
2𝑠
2=𝐵𝐶
2
𝐵𝐶
=𝑠 √2
,
Whi
ch is
wha
t we
wan
ted
to s
how
.
Figu
re 2
3
The
tick
mar
ks o
n th
e tw
o le
gs in
dica
te th
at th
ose
two
lines
are
the
sam
e le
ngth
.
§1
Rig
ht tr
iang
les
– 87
Exam
ple
2
Det
erm
ine
the
leng
ths
of th
e m
issin
g sid
es in
Fig
ure
24.
Appl
ying
the
spec
ial t
riang
le ra
tio, w
e ha
ve th
at 𝑠
=1
5. T
hus
the
othe
r leg
,
𝑌𝑍
=1
5,
and
the
hypo
tenu
se,
𝑋𝑍
=1
5√2
.
Keep
in m
ind
Pyth
agor
as’s
Theo
rem
can
stil
l be
used
to d
eter
min
e th
e pr
evio
us tr
iang
le,
alth
ough
you
wou
ld h
ave
to a
pply
the
fact
that
𝑋𝑌
=𝑌𝑍
.
Let’s
take
a lo
ok a
t one
mor
e sp
ecia
l rig
ht tr
iang
le ra
tio.
30
°−
60
°−
90
° Rig
ht tr
iang
le ra
tio
Giv
en a
righ
t tria
ngle
with
ang
les
of 3
0°,
60
°, a
nd 9
0°,
then
the
shor
t leg
has
leng
th 𝑠
, th
e lo
nger
leg
has
a le
ngth
of 𝑠√
3, a
nd th
e hy
pote
nuse
has
a le
ngth
of 2𝑠.
W
e w
ill p
rove
the
prev
ious
resu
lt, b
ut o
ur p
roof
will
be
a bi
t mes
sy. W
e’ll
look
for a
bet
ter
way
to p
rove
it a
s w
e co
ntin
ue th
roug
h th
is U
nit.
Proo
f.
To fu
lly p
rove
this
asse
rtio
n, w
e w
ill n
eed
to p
rove
thre
e di
ffere
nt p
ossib
ilitie
s:
Figu
re 2
4
Uni
t fou
r – 8
8
I. A
right
tria
ngle
with
legs
of 𝑠
and
𝑠√
3,
II.
A rig
ht tr
iang
le w
ith a
leg
of 𝑠
and
a h
ypot
enus
e of
2𝑠,
and
III
. A
right
tria
ngle
with
a le
g of
𝑠√
3 a
nd a
hyp
oten
use
of 2𝑠.
We
will
pro
ve th
e fir
st p
oint
, the
n le
ave
the
final
two
proo
fs to
the
read
er.
We
are
give
n a
right
tria
ngle
with
leng
ths
of 𝑠
and
𝑠√
3. T
hus,
usin
g Py
thag
oras
, we
have
𝑎=𝑠,𝑏
=𝑠 √
3
whe
nce
𝑠2
+(𝑠√
3)2
=𝑐
2.
Cons
eque
ntly
, we
get t
hat
𝑠2
+3𝑠
2=𝑐
2
4𝑠
2=𝑐
2
𝑐=2𝑠,
whi
ch is
wha
t we
wan
ted
to s
how
.
Exam
ple
3
Com
plet
e th
e rig
ht tr
iang
le s
how
n in
Fig
ure
25.
To “
com
plet
e” a
rig
ht t
riang
le m
eans
to
dete
rmin
e al
l of
its
sid
e le
ngth
s an
d an
gles
m
easu
res.
In o
ur c
ase,
we
mus
t det
erm
ine
one
angl
e m
easu
re a
nd th
e le
ngth
of b
oth
legs
. W
e be
gin
with
the
angl
e m
easu
re. R
ecal
l fro
m G
eom
etry
that
the
sum
of t
hree
ang
les
in
a tr
iang
le is
alw
ays
18
0°.
Thus
our
miss
ing
angl
e (w
hich
we’
ll ca
ll 𝛼
)iii is
𝛼=
18
0°−( 3
0°−
90
°)
iii It
is m
y cu
stom
to n
ame
side
s w
ith lo
wer
-cas
e La
tin le
tter
s, 𝑎
,𝑏,𝑐
, and
then
cal
l the
ir op
posit
e an
gles
w
ith lo
wer
-cas
e G
reek
lett
ers, 𝛼
,𝛽,𝛾
, res
pect
ivel
y. T
his
is m
y ow
n pe
rson
al c
onve
ntio
n, a
nd I
wel
com
e yo
u to
dev
elop
you
r ow
n.
Figu
re 2
5
§1
Rig
ht tr
iang
les
– 89
𝛼=
60
°.
We’
ll us
e th
e ra
tios
show
n in
our
pre
viou
s re
sult
to d
eter
min
e th
e sid
e le
ngth
s. Re
call
from
Geo
met
ry th
at th
at th
e sh
orte
st si
de o
f a tr
iang
le w
ill a
lway
s be
oppo
site
the
shor
test
an
gle.
Sin
ce, i
n a
30
°−
60
°−
90
° tr
iang
le, t
he h
ypot
enus
e is
alw
ays
twic
e th
e le
ngth
of
the
shor
test
sid
e (w
hich
we’
ll ca
ll 𝑏
), w
e ca
n w
rite
14
=2𝑏
,
And
henc
e
𝑏=
7.
To d
eter
min
e th
e on
e re
mai
ning
sid
e, w
hich
is th
e lo
nger
leg
(sin
ce it
is o
ppos
ite th
e 6
0°
angl
e), w
e si
mpl
y m
ultip
ly o
ur p
revi
ous
resu
lt by
√3
. Hen
ce o
ur r
emai
ning
sid
e (w
hich
w
e’ll
call 𝑎
) is
𝑎=
7√
3.
We
show
our
fina
l res
ults
in F
igur
e 26
.
Pyth
agor
as’ T
heor
em a
llow
s us
to d
eter
min
e th
e th
ird s
ide
of a
righ
t tria
ngle
whe
n on
ly
give
n tw
o ot
her
sides
. Thi
s is
very
use
ful,
but
it le
aves
out
som
e ot
her
info
rmat
ion.
For
ex
ampl
e, w
hat
if w
e w
ante
d to
det
erm
ine
the
angl
e m
easu
res
whe
n gi
ven
two
side
leng
ths?
Unl
ess w
e m
easu
re it
, we
wou
ldn’
t be
able
to te
ll.iv O
r, w
hat i
f we
had
only
kno
wn
one
of th
e sid
es o
f our
tria
ngle
? Th
en w
e co
uld
not d
eter
min
e an
ythi
ng e
lse.
This
is o
ne o
f the
rea
sons
tha
t th
ese
spec
ial r
ight
tria
ngle
s ar
e so
use
ful.
We’
ll us
e th
at
fact
a m
yria
d of
tim
es in
this
Part
– a
nd w
e’ll
com
e ba
ck to
it v
ery
shor
tly. D
oubt
less
, of
cour
se, y
ou a
lso s
ee th
e ad
vant
age
to s
olvi
ng a
righ
t tria
ngle
as
quic
kly
and
effic
ient
ly a
s w
e ju
st d
emon
stra
ted.
iv A
nd e
ven
if w
e di
d m
easu
re it
, we
wou
ld h
ave
to d
raw
our
tria
ngle
per
fect
ly to
sca
le, a
nd e
ven
then
, th
ere
is a
deg
ree
of e
rror
in m
easu
rem
ent.
Figu
re 2
6
Uni
t fou
r – 9
0
Allo
w u
s to
reite
rate
the
impo
rtan
ce o
f thi
s se
ctio
n: Y
ou s
impl
y m
ust b
ecom
e ve
ry a
dept
at
sol
ving
spe
cial
righ
t tria
ngle
s. W
e ca
nnot
ove
rem
phas
ize
how
foun
datio
nal t
his
is. Y
ou
will
util
ize
this
tool
aga
in a
nd a
gain
in th
e fo
rthc
omin
g se
ctio
ns. W
e’ve
giv
en y
ou m
any
prac
tice
prob
lem
s in
thi
s se
ctio
n, b
ut y
ou w
ill n
eed
to t
ake
extr
a ca
re t
o m
aste
r it
for
your
self.
§𝟏 E
xerc
ises
1.)
In th
e fo
llow
ing
prob
lem
s, as
sum
e th
at 𝑎
and
𝑏 a
re th
e le
ngth
s of
legs
, and
that
𝑐
is th
e le
ngth
of a
hyp
oten
use.
Fin
d th
e le
ngth
of t
he m
issin
g sid
e.
(A) 𝑎
=12
,𝑏=
15
(B
) 𝑎
=1
.5,𝑏
=2
.3
(C) 𝑎
=24
,𝑏=
8
(D) 𝑎
=12
.5,𝑐
=1
9
(E)
𝑏=
11
3,𝑐
=2
01
(F
) 𝑐
=4
.5,𝑎
=1
(G
) 𝑐
=2
7,𝑏
=24
(H
) 𝑎
=1
0,𝑏
=1
2.
) As
sum
e yo
u ha
ve a
tria
ngle
with
leg
leng
ths
of 5
and
7.
(A)
Troy
lets
𝑎=
5 a
nd 𝑏
=7
. Tin
a le
ts 𝑎
=7
and
𝑏=
5. D
o th
ey g
et th
e sa
me
resu
lt fo
r 𝑐?
(B)
Prov
e th
at 𝑎
and
𝑏 a
re in
terc
hang
eabl
e w
hen
usin
g Py
thag
oras
’ The
orem
. 3.
) Ar
e al
l tria
ngle
s po
ssib
le?
For
exam
ple,
can
you
hav
e a
right
tria
ngle
with
legs
of
leng
th 1
0 a
nd 2
0 a
nd a
hyp
oten
use
of 1
5?
Why
or w
hy n
ot?
4.)
Reca
ll th
at a
ll tr
iang
les’
angl
es a
dd u
p to
18
0°.
With
thi
s in
min
d, c
ompl
ete
the
follo
win
g rig
ht tr
iang
les.
(A)
(B)
(C)
(D)
5.
) So
far,
we’
ve s
een
that
𝑎2
+𝑏
2=𝑐
2 is
tru
e fo
r al
l rea
l num
bers
– t
hat
is, a
t le
ast
one
of t
he t
hree
len
gths
was
irr
atio
nal.
Is i
t po
ssib
le t
hat
all
thre
e le
ngth
s ar
e in
tege
rs?
Let’s
exp
lore
…
(A) 𝑎
=3
,𝑏=4
,𝑐=
?
§1
Rig
ht tr
iang
les
– 91
(B)
𝑎=
?,𝑏
=12
,𝑐=
13
(C
) Se
eif
you
can
disc
over
thre
e ot
her P
ytha
gore
anTr
iple
s.6.
) Th
e fo
llow
ing
tabl
e re
pres
ents
a s
mal
l lis
t of
Pyt
hago
rean
Trip
les.
(Eac
h ro
w
cont
ains
the
thre
e sid
e le
ngth
s of
the
tria
ngle
.) Si
de le
ngth
s 3
4
5
Si
de le
ngth
s 6
8
1
0
Side
leng
ths
9
12
1
5
(A)
Wha
t pat
tern
exi
sts
in th
is ta
ble?
(B
) U
se th
e pa
tter
n to
list
out
thre
e m
ore
Pyth
agor
ean
Trip
les.
(C)
How
man
y Py
thag
orea
n Tr
iple
s ex
ist?
(D)
Thes
e w
ere
all m
ultip
les
of t
he s
mal
lest
(and
mos
t co
mm
on) P
ytha
gore
an
Trip
le, t
he 3
,4,5
tria
ngle
.v Will
this
tric
k w
ork
with
oth
er d
istin
ct P
ytha
gore
an
Trip
les,
like
the
5,12
,13
tria
ngle
? 7.
) Co
mpl
ete
the
follo
win
g rig
ht tr
iang
les.
(A)
(B)
(C)
(F)
(G)
v W
e hi
ghly
enc
oura
ge y
ou to
mem
oriz
e th
e 3
,4,5
tria
ngle
. It p
ops
up e
noug
h th
at m
emor
izin
g it
will
sav
e yo
u so
me
cons
ider
able
tim
e.
Uni
t fou
r – 9
2 (D)
(E)
(H)
8.)
As a
mat
ter
of c
onve
ntio
n, w
e of
ten
rew
rite
any
ratio
s th
at h
ave
an i
rrat
iona
l nu
mbe
r in
the
ir de
nom
inat
or s
uch
that
the
num
erat
or i
s en
tirel
y ra
tiona
l. Fo
r ex
ampl
e,
1 √2
=1 √2∙√2
√2
=√2 2
.
In m
ost c
ases
, thi
s is
pur
ely
by c
onve
ntio
n. O
ne re
ason
for d
oing
this
is th
at y
ou
will
onl
y ne
ed to
mem
oriz
e on
e se
t of n
umbe
rs w
hen
it co
mes
to s
peci
fic T
rig
ratio
s.
(A)
Wha
t is
the
den
omin
ator
in t
he e
xam
ple
show
n ab
ove?
And
wha
t is
the
nu
mer
ator
and
den
omin
ator
of t
he n
umbe
r mul
tiplie
d by
1 √2?
(B)
Wha
t is √
2
√2?
(C)
Base
d on
you
r an
swer
to
(B),
has
the
valu
e of
1 √2 c
hang
ed a
fter
the
mul
tiplic
atio
n? U
se a
cal
cula
tor i
f you
’re n
ot c
onvi
nced
. (D
) U
se th
e ab
ove
exam
ple
to ra
tiona
lize
the
follo
win
g de
nom
inat
ors.
i. 3 √3
ii.
4 √2
iii.
√2
√3
iv.
𝜋 √7
v.
3 √4
vi.
5
2√
2
vii.
24
√1
2
viii.
√3
√10
9.)
Answ
er T
rue
or F
alse
. (A
) A 4
5°−4
5°−
90
° tr
iang
le c
an n
ever
hav
e th
ree
side
leng
ths
whi
ch a
re a
ll w
hole
num
bers
.
§1
Rig
ht tr
iang
les
– 93
(B)
The
shor
test
sid
e of
a tr
iang
le is
alw
ays
oppo
site
of th
e sh
orte
st a
ngle
. (C
) Th
e hy
pote
nuse
is a
lway
s th
e lo
nges
t sid
e of
a tr
iang
le.
(D)
A tr
iang
le c
an n
ever
hav
e m
ore
than
one
90
° ang
le.
(E)
Ther
e is
an
infin
ite a
mou
nt o
f Pyt
hago
rean
Trip
les.
10.)
In y
our o
wn
wor
ds, d
escr
ibe
the
shor
tcut
to c
ompl
etin
g an
isos
cele
s rig
ht tr
iang
le.
11.)
In y
our o
wn
wor
ds, d
escr
ibe
the
shor
tcut
to c
ompl
etin
g a
30
°−
60
°−
90
° tria
ngle
. 12
.) W
ill y
ou fo
rget
thes
e sp
ecia
l tria
ngle
ratio
s?
13.)
Whe
re w
ill y
ou lo
ok if
you
forg
et th
em?
14.)
You’
re s
ure
you
won
’t fo
rget
them
??
§2
Th
e Si
ne, C
osin
e, a
nd T
ange
nt fu
nctio
ns
We
used
Geo
met
ry in
the
pre
viou
s se
ctio
n (a
nd w
e w
ill fr
eque
ntly
fall
back
on
it) a
nd it
ha
d so
me
good
use
s, bu
t in
this
sect
ion
we
will
dis
cove
r a n
ew s
et o
f too
ls th
at w
ill a
llow
us
to d
o ev
en m
ore.
Mor
e im
port
antly
, tho
ugh,
is th
at w
e’ll
be a
ble
to re
late
ang
le m
easu
res
and
side
leng
ths.
Cert
ainl
y, t
his
will
allo
w u
s to
cal
cula
te m
ore
than
we
coul
d pr
evio
usly
, but
we
will
also
fin
d a
shre
d of
trut
h th
at w
e fe
lt w
as th
ere
but c
ould
n’t q
uite
asc
erta
in.
To b
egin
with
, we’
ll de
fine
a fu
nctio
n th
at ta
kes
an a
cute
ang
le a
s its
inpu
t and
retu
rns
a ra
tio o
f tw
o sid
e le
ngth
s.
The
Sine
func
tion
Th
e Si
ne fu
nctio
n ac
cept
s as a
n in
put a
n ac
ute
angl
e of
a ri
ght t
riang
le, a
nd th
en re
turn
s th
e ra
tio o
f the
sid
e op
posit
e to
the
angl
e to
the
hypo
tenu
se. S
ymbo
lical
ly,
sin𝛼
=𝑎 𝑐
.
We
coul
d al
so s
tate
that
sin𝛽
=𝑏 𝑐
.
Not
e th
at o
ur c
urre
nt d
efin
ition
onl
y ac
cept
s ac
ute
angl
es o
f a ri
ght t
riang
le.
Uni
t fou
r – 9
4
Exam
ple
1a
Writ
e th
e ra
tio o
f the
sin
e of
ang
le 𝛼
and
𝛽 g
iven
Fig
ure
27.
Let’s
sta
rt w
ith th
e sin
e of
ang
le 𝛼
, whi
ch is
den
oted
as
sin𝛼
. Thi
s is
just
a ra
tio –
so
all w
e ne
ed to
do
is cr
eate
a fr
actio
n. S
ince
we’
ve d
efin
ed th
is fu
nctio
n as
the
side
oppo
site
to
the
hypo
tenu
se, w
e ju
st n
eed
to w
rite
sin𝛼
=3 6.7
,
Sinc
e 3
is th
e sid
e op
posit
e an
d 6
.7 is
the
hypo
tenu
se. I
t rea
lly is
that
sim
ple!
The
sine
of a
ngle
𝛽 is
han
dled
the
sam
e w
ay, e
xcep
t no
tice
that
the
sid
e op
posit
e 𝛽
is
diffe
rent
than
the
side
oppo
site
of 𝛼
. In
this
case
we
have
sin𝛽
=6 6.7
.
Wha
teve
r ang
le 𝛼
is, w
hen
we
inpu
t it i
nto
the
sine
func
tion,
we
rece
ive
as a
n ou
tput
3 6.7
.
The
sam
e is
true
for 𝛽
; whe
n w
e in
put 𝛽
into
the
sine
func
tion,
we
rece
ive
as a
n ou
tput
6 6.7
.
Of c
ours
e, th
at h
asn’
t rea
lly re
veal
ed to
us
any
new
info
rmat
ion.
Let
’s tr
y a
prob
lem
that
te
ache
s us
som
ethi
ng, s
hall
we?
Exam
ple
1b
Eval
uate
sin
30
° giv
en F
igur
e 28
.
Figu
re 2
7
Figu
re 2
8
§2
The
Sin
e, C
osin
e, a
nd T
ange
nt fu
nctio
ns –
95
To a
nsw
er th
is, w
e sim
ply
need
to s
et u
p ou
r rat
io. B
ased
on
wha
t we’
ve b
een
give
n, w
e ha
ve
sin
30
°=
2
hyp
ote
nu
se .
But
wha
t is
the
leng
th o
f th
ehy
pote
nuse
? W
e ca
n ea
sily
find
that
out
usi
ng w
hat
we
lear
ned
in th
e pr
evio
us se
ctio
n.i S
ince
2 is
the
leng
th o
f the
shor
t leg
, the
n th
e hy
pote
nuse
m
ust
be t
wic
e th
at le
ngth
. Hen
ce w
e co
nclu
de t
hat
they
hyp
oten
use
has
a le
ngth
of 4
. Th
eref
ore
sin
30
°=2 4
=1 2
.
So w
hat d
id w
e fin
d ou
t? W
ell,
if yo
u in
put 3
0° i
nto
the
sine
func
tion,
you
sho
uld
get o
ut
1 2. B
ut th
ere’
s a
very
impo
rtan
t que
stio
n to
ans
wer
: W
ill th
is on
ly h
appe
n w
hen
the
shor
t
leg
of a
30
°−
60
°−
90
° tr
iang
le h
as a
leng
th o
f 2?
Put a
noth
er w
ay, d
oes
the
size
of t
he
tria
ngle
pla
y a
role
in d
eter
min
ing
the
resu
lt of
sin
30
°?
To a
nsw
er t
his
ques
tion,
we
will
do
as a
ll m
athe
mat
icia
ns d
o:
We
will
exp
lore
, we
will
ex
perim
ent,
we
will
pla
y! W
e’ll
mak
e a
new
tria
ngle
with
an
angl
e of
30
° tha
t has
a d
iffer
ent
size
and
obse
rve
wha
t hap
pens
.
Exam
ple
1c
Eval
uate
sin
30
° usi
ng F
igur
e 29
.
i T
his
is e
xact
ly w
hat w
e m
ean
we
say
that
you
will
nee
d to
be
very
ade
pt a
t the
pre
viou
s se
ctio
n. T
his
is
ofte
n ho
w w
e’ll
calc
ulat
e ou
r Trig
func
tions
– b
y se
ttin
g up
a s
peci
al ri
ght t
riang
le, a
nd fi
lling
in th
e re
st o
f th
e in
form
atio
n. T
here
fore
, we’
ll st
ate
it ag
ain:
Lea
rn th
e pr
evio
us s
ectio
n!
Figu
re 2
9
Uni
t fou
r – 9
6
We
appl
y th
e sa
me
proc
edur
e. In
thi
s ca
se, t
he h
ypot
enus
e m
ust
be 2
0. S
ince
the
sin
e ra
tio is
opposite
hypotenuse
, and
the
side
leng
th o
ppos
ite to
30
° is
10
, we
conc
lude
that
sin
30
°=
10
20
=1 2
.
Now
wai
t jus
t a
min
ute,
tha
t’s v
ery
inte
rest
ing!
Why
did
a d
iffer
ent
size
tria
ngle
giv
e us
th
e sa
me
resu
lt? T
his
is m
ost u
nexp
ecte
d!
Inde
ed, m
uch
of m
athe
mat
ics
was
disc
over
ed in
the
sam
e w
ay. I
t st
arts
with
a s
impl
e cu
riosit
y, a
nd th
en a
con
scio
us e
ffort
to p
oke
and
prod
wha
teve
r you
hav
e un
til it
giv
es
up s
ome
prof
ound
trut
h.ii T
hink
of m
ath
like
a pi
ñata
, whi
ch c
onta
ins
so m
uch
delic
ious
tr
uth.
But
the
onl
y w
ay t
o ge
t at
it is
to
swin
g at
it a
few
tim
es (
and
ofte
ntim
es t
his
swin
ging
is tr
uly
diffi
cult!
).
In th
is ca
se, t
he p
rofo
und
trut
h se
ems
to b
e th
at s
in3
0°
=1 2 n
o m
atte
r wha
t siz
e tr
iang
le
we
have
. Thu
s w
e sh
are
the
follo
win
g ve
ry im
port
ant t
heor
em.
The
depe
nden
cy o
f sin𝛼
Th
e ou
tput
of s
in𝛼
onl
y de
pend
s on
𝛼. T
he s
ize
of th
e tr
iang
le is
irre
leva
nt.
Ther
e is
, of c
ours
e, a
rea
son
that
sin𝛼
doe
s no
t dep
end
on th
e si
ze o
f the
tria
ngle
, and
w
e’ll
brie
fly e
xplo
re th
is in
the
Exer
cise
s.
This
prof
ound
trut
h w
ill a
llow
us
to c
onst
ruct
any
siz
e tr
iang
le w
e w
ish
whe
n w
e go
abo
ut
dete
rmin
ing
the
sine
of a
ny a
ngle
. Thi
s is
use
ful,
since
som
e ch
oice
s w
ill b
e ea
sier
tha
n ot
hers
. In
fact
, it i
s th
is v
ery
fact
that
will
mot
ivat
e ou
r wor
k in
the
next
uni
t. Le
t’s b
riefly
ex
plor
e th
is id
ea.
Exam
ple
1d
Eval
uate
sin4
5°.
Base
d on
wha
t w
e ju
st le
arne
d, w
e sh
ould
cre
ate
a 4
5°−4
5°−
90
° tr
iang
le, a
nd t
hen
crea
te a
ratio
usi
ng th
e le
ngth
s of
the
oppo
site
side
and
the
hypo
tenu
se. I
t wou
ld b
e fa
ir to
ask
wha
t tria
ngle
, exa
ctly
, you
sho
uld
crea
te. W
e st
art w
ith th
e ba
sics
in F
igur
e 30
.
ii A
nd th
is pr
oces
s of
exp
erim
enta
tion
and
verif
icat
ion
can
be a
s qu
ick
and
pain
less
as
we
just
witn
esse
d,
or it
cou
ld ta
ke s
ever
al y
ears
. Or c
entu
ries.
But i
t’s th
e jo
urne
y th
at c
ount
s, rig
ht?
§2
The
Sin
e, C
osin
e, a
nd T
ange
nt fu
nctio
ns –
97
We’
ve d
raw
n so
me
isosc
eles
righ
t tria
ngle
. Now
how
sho
uld
we
labe
l the
ang
les
and
the
side
leng
ths?
Cer
tain
ly, t
he tw
o no
n-rig
ht a
ngle
s m
ust b
e 4
5°
(sin
ce th
ey a
re b
oth
equa
l an
d m
ust e
qual
90
°), b
ut w
hat a
bout
the
side
leng
ths?
We
coul
d, fo
r exa
mpl
e, s
tart
with
th
e le
gs, a
nd la
bel e
ach,
say
, 10
. Onc
e w
e’ve
mad
e th
at d
ecisi
on, w
e m
ust c
oncl
ude
that
th
e hy
pote
nuse
is 1
0√2
.iii W
e dr
aw th
is in
Fig
ure
31.
Ther
e’s
noth
ing
wro
ng w
ith t
he p
revi
ous
pict
ure,
but
it
also
isn
’t th
e be
st. S
houl
d w
e ch
oose
a s
mal
l num
ber?
Or,
shou
ldn’
t we
choo
se s
ome
num
ber w
here
we
won
’t ha
ve a
ny
simpl
ifica
tion?
Beca
use
of th
is, w
hy d
on’t
we
let t
he le
gs b
e of
leng
th 1
.iv T
hen
we
have
Fig
ure
32.
iii S
ee th
e pr
evio
us s
ectio
n.
iv Y
ou m
ay n
ot s
ee th
at th
is is
the
“sim
ples
t” n
umbe
r. Th
at’s
fine!
Thi
s is
som
ethi
ng y
ou fi
nd o
ut th
roug
h ex
perim
enta
tion
and
expl
orat
ion.
Figu
re 3
0
Figu
re 3
1
Uni
t fou
r – 9
8
To c
alcu
late
sin4
5°,
then
, we
just
nee
d to
set
up
our r
atio
acc
ordi
ngly
.v We
have
sin4
5°
=1 √2
.
Wha
t if w
e ha
d ch
osen
the
orig
inal
tria
ngle
, whe
re th
e le
gs h
ad a
leng
th o
f 10
? Th
en w
e w
ould
hav
e ha
d
sin4
5°
=1
0
10√2
=1 √2
,
Whi
ch is
the
sam
e an
swer
, alth
ough
it w
as o
ne le
ss s
tep
to g
et to
the
ans
wer
. In
futu
re
sect
ions
, we’
ll be
pic
king
a v
ery
part
icul
ar tr
iang
le to
wor
k w
ith w
hen
we
wan
t to
eval
uate
so
me
Trig
func
tion.
We
are
just
bei
ng c
leve
r, an
d tr
ying
to
save
our
selv
es s
ome
time
– al
thou
gh th
is is
not m
anda
tory
, and
you
are
free
to p
ick
any
tria
ngle
you
like
.vi
A sin
gle
step
may
see
m in
sign
ifica
nt, b
ut s
ince
we’
ll be
goi
ng b
ack
to th
ese
two
spec
ial
tria
ngle
s ag
ain
and
agai
n,vi
i you
’ll w
ant
to s
trea
mlin
e th
e pr
oces
s as
muc
h as
pos
sible
. Re
gard
less
, we
now
hav
e an
othe
r va
lue
or r
atio
to
add
to o
ur li
st, w
hich
you
’ll w
ork
on
com
plet
ing
in th
e Ex
erci
ses.
One
mor
e no
te b
efor
e w
e co
ntin
ue o
n: M
ost t
extb
ooks
will
tell
you
that
sin4
5°
=√
2 2. A
re
they
wro
ng?
And
whe
re d
id t
hey
get
that
ans
wer
fro
m?
And
why
wou
ld t
hey
both
er
writ
ing
it lik
e th
at?
Goo
d qu
estio
ns!
Let u
s now
mov
e on
to th
e Co
sine
func
tion.
By
its v
ery
nam
e, w
e ca
n in
fer t
hat i
t is r
elat
ed
to th
e sin
e fu
nctio
n, a
nd –
as
we’
ll so
on s
ee –
this
is v
ery
muc
h th
e ca
se.
v W
ill it
mat
ter w
hich
45
° ang
le w
e ch
oose
? Se
e fo
r you
rsel
f! vi W
e w
ill re
itera
te th
is po
int i
n fu
ture
sec
tions
. vi
i And
aga
in!
Figu
re 3
2
§2
The
Sin
e, C
osin
e, a
nd T
ange
nt fu
nctio
ns –
99
The
Cosin
e fu
nctio
n
Th
e Co
sine
func
tion
acce
pts
as a
n in
put
an a
cute
ang
le o
f a
right
tria
ngle
, and
the
n re
turn
s th
e ra
tio o
f the
sid
e ad
jace
nt to
the
angl
e to
the
hypo
tenu
se. S
ymbo
lical
ly,
cos𝛼
=𝑏 𝑐
.
We
coul
d al
so s
tate
that
co
s𝛽
=𝑎 𝑐
.
The
Cosin
e fu
nctio
n is
ver
y sim
ilar
to t
he S
ine
func
tion.
It, t
oo, a
ccep
ts a
n an
gle
as a
n in
put.
But
this
time
it re
turn
s as
an
outp
ut t
he s
ide
leng
th a
djac
entvi
ii to
𝛼 o
ver
the
hypo
tenu
se. T
his
– ag
ain
– ha
s on
ly b
een
defin
ed fo
r acu
te a
ngle
s in
a ri
ght t
riang
le.
Exam
ple
2a
Writ
e th
e Co
sine
ratio
of 𝛼
giv
en F
igur
e 33
.
All w
e ne
ed to
do
is m
ake
a fra
ctio
n of
the
leng
th o
f the
adj
acen
t leg
ove
r the
hyp
oten
use.
Th
is is
sim
ple:
We
get
vi
ii Not
e th
at a
djac
ent m
eans
touc
hing
. And
whi
le, y
es, t
he h
ypot
enus
e is
touc
hing
𝛼, w
hat w
e re
ally
mea
n is
the
adja
cent
leg.
Figu
re 3
3
Uni
t fou
r – 1
00
cos𝛼
=4 5
.
It is
impo
rtan
t to
note
that
thes
e ra
tios
will
onl
y w
ork
with
righ
t tria
ngle
s. W
as th
e ab
ove
tria
ngle
a ri
ght t
riang
le?
It m
ight
look
that
way
, but
how
can
we
be su
re?ix
Up
to th
is po
int,
we’
ve a
lway
s se
en t
he b
ox a
t th
e rig
ht a
ngle
. Her
e w
e do
n’t
have
one
– s
o is
our
res
ult
inco
rrec
t?
This
is a
n ex
celle
nt q
uest
ion
and
one
you
mus
t be
awar
e of
. Pyt
hago
ras’
Theo
rem
tells
us
that
for a
ny ri
ght t
riang
le,
𝑎2
+𝑏
2=𝑐
2.
The
conv
erse
of
Pyth
agor
as’ T
heor
em –
tha
t is,
if 𝑎
2+𝑏
2=𝑐
2 t
hen
you
have
a r
ight
tr
iang
le –
is a
lso tr
ue. S
ince
32
+4
2=
52
,
we
can
conc
lude
we
have
a r
ight
tria
ngle
. Th
us t
he a
fore
men
tione
d co
nclu
sion
(that
co
s𝛼
=4 5) i
s co
rrec
t.
Exam
ple
2b
List
out
the
Sine
and
Cos
ine
ratio
s fo
r ang
les 𝛼
and
𝛽 in
Fig
ure
34.
Rem
embe
r to
first
ver
ify th
at y
ou h
ave
a rig
ht tr
iang
le. S
ince
52
+12
2=
13
2,
we
can
mov
e on
to th
e ne
xt s
tep.
ix F
or e
xam
ple,
if th
e un
nam
ed a
ngle
, cal
l it 𝛾
, is
equa
l to
89
.99
(whi
ch is
real
ly c
lose
to 9
0) t
hen
our T
rig
ratio
s ar
e in
accu
rate
. We
mus
tbe
posi
tive
we
have
an
actu
al ri
ght t
riang
le; i
t is
not g
ood
enou
gh to
al
mos
t be
a rig
ht tr
iang
le.
Figu
re 3
4
§2
The
Sin
e, C
osin
e, a
nd T
ange
nt fu
nctio
ns –
101
Let u
s de
al w
ith 𝛼
firs
t. O
ur ra
tios
for 𝛼
are
sin𝛼
=5 13
and
cos𝛼
=12
13
.
Then
, with
resp
ect t
o 𝛽
, we
have
sin𝛽
=12
13
and
cos𝛽
=5 13
.
But i
sn’t
ther
e so
met
hing
pec
ulia
r abo
ut o
ur re
sults
? Lo
ok a
gain
– is
ther
e an
ythi
ng th
at
you
can
see?
Why
is it
that
sin𝛼
=5 13
=co
s𝛽
?
Or t
hat
sin𝛽
=12
13
=co
s𝛼
?
This
is a
not
e-w
orth
y fin
d! W
hy s
houl
d tw
o di
ffere
nt f
unct
ions
(w
ith d
iffer
ent
inpu
ts)
prod
uce
the
sam
e ou
tput
?
Befo
re w
e fo
rmal
ize
our n
ext t
heor
em, l
et u
s con
sider
the
rela
tions
hip
that
exi
sts b
etw
een
the
two
non-
right
ang
les
in a
righ
t tria
ngle
.
Firs
t, le
t us
(in F
igur
e 35
) dra
w s
ome
right
tria
ngle
.
Fi
gure
35
Uni
t fou
r – 1
02
Let
us la
bel t
he t
wo
non-
right
ang
les
as 𝛼
and
𝛽. I
s th
ere
som
e re
latio
nshi
p be
twee
n th
em?
We
now
nee
d so
me
sort
of s
tart
ing
poin
t –
so le
t’s t
hink
bac
k to
oth
er r
elat
ions
hips
in
tria
ngle
s. Th
e bi
gges
t on
e, p
erha
ps, i
s th
e th
eore
m th
at th
e su
m o
f all
thre
e an
gles
in a
tr
iang
le a
re 1
80
°. So
let’s
beg
in w
ith th
is, p
lay
with
it a
bit,
and
then
see
if w
e ca
n’t u
ncov
er
som
ethi
ng o
f not
e.x
Usin
g th
e pr
evio
us re
latio
nshi
p, w
e w
ould
then
hav
e
𝛼+𝛽
+𝛾
=1
80
.
But w
e kn
ow 𝛾
=9
0°,
since
it is
a ri
ght a
ngle
(and
by
defin
ition
, all
right
ang
les e
qual
90
°).
Subs
titut
ing
this
in, w
e se
e th
at
𝛼+𝛽
=9
0°.
This
yiel
ds u
s on
e im
port
ant
rela
tions
hip:
𝛼
and
𝛽 a
re c
ompl
emen
tary
. Th
is se
ems
beni
gn…
But
let u
s co
nsid
er th
e w
ord
Cosin
e fo
r a m
omen
t. It
is ac
tual
ly a
shor
tene
d fo
rm
of th
e te
rm “
Com
plem
ent
of t
he s
ine.
” W
ait a
mom
ent…
Is it
a c
oinc
iden
ce th
at w
e ju
st
show
ed th
at 𝛼
and
𝛽 a
re c
ompl
emen
tary
and
the
Cosin
e fu
nctio
n ha
s in
its v
ery
nam
e th
e w
ord
“com
plem
ent”
? M
ost
cert
ainl
y no
t!xi W
e’ve
disc
over
ed s
ome
prof
ound
tru
th!
We
now
pro
vide
a fo
rmal
pro
of.
Proo
f.
We
wan
t to
sh
ow
that
th
e tw
o no
n-rig
ht
angl
es
of
a rig
ht
tria
ngle
m
ust
be
com
plem
enta
ry, t
hat i
s, th
at th
eir s
um m
ust b
e 9
0°.
Cons
truc
t rig
ht Δ𝐴𝐵𝐶
with
righ
t ang
le 𝐵
.
x O
f cou
rse,
our
exp
lora
tion
may
turn
out
frui
tless
. Tha
t’s O
K! B
elie
ve it
or n
ot, f
indi
ng o
ut w
hat d
oesn
’t w
ork
can
also
ver
y he
lpfu
l. Ad
ditio
nally
, don
’t ge
t dis
cour
aged
if y
ou’re
not
cor
rect
the
first
tim
e –
very
fe
w m
athe
mat
icia
ns a
re!
xi W
hich
doe
sn’t
mea
n “N
ever
”. It
just
mea
ns u
nlik
ely.
Som
e m
ay s
coff
at th
e si
mpl
e lo
gic
we
used
her
e,
but k
eep
in m
ind
we’
re s
how
ing
a pr
oces
s of
com
ing
to a
n in
form
al c
oncl
usio
n. Y
ou m
ay w
ish th
e ex
plan
atio
n w
as m
ore
form
al, a
nd if
that
’s th
e ca
se, I
com
men
d yo
u an
d en
cour
age
you
to w
rite
your
ow
n te
xtbo
ok.
§2
The
Sin
e, C
osin
e, a
nd T
ange
nt fu
nctio
ns –
103
Beca
use
the
sum
of a
ll th
ree
angl
es in
a tr
iang
le is
18
0°,
we
know
that
𝐴+𝐵
+𝐶
=1
80
°. Bu
t 𝐵=
90
°, sin
ce (b
y de
finiti
on) a
ll rig
ht a
ngle
s ar
e eq
ual t
o 9
0°.
Then
, by
subs
titut
ion,
w
e ha
ve th
at 𝐴
+9
0°
+𝐶
=1
80
°. U
sing
the
subt
ract
ion
prop
erty
of e
qual
ity re
veal
s th
at
𝐴+𝐶
=9
0°,
whi
ch is
pre
cise
ly w
hat t
he w
ord
“com
plem
enta
ry” m
eans
.
We
have
thus
sho
wn
wha
t we
wan
ted
to s
how
.
With
this
in m
ind,
let u
s fo
rmal
ize
the
prev
ious
.
The
rela
tions
hip
betw
een
the
Sine
and
Cos
ine
func
tions
. Th
e tw
o no
n-rig
ht a
ngle
s in
a ri
ght t
riang
le a
re a
lway
s co
mpl
emen
tary
. Ex
ampl
e 2c
Eval
uate
co
s3
0°.
We
go th
roug
h th
e sa
me
proc
edur
e fo
und
in E
xam
ples
1c
and
1d. W
e fir
st d
raw
a ri
ght
tria
ngle
xii (
Figu
re 3
6).
Base
d on
the
tria
ngle
that
we
chos
e, w
e ha
ve th
at
cos𝛼
=ad
jace
nt
1.
xi
iRe
mem
ber t
hat w
e ar
e fr
ee to
dra
w a
ny ri
ght t
riang
le w
e w
ant.
Wel
l, as
long
as
is it
has
a 3
0° a
ngle
, an
yway
.
Figu
re 3
6
We
chos
e to
mak
e ou
r hyp
oten
use
1 h
ere.
The
re a
re a
few
oth
er n
ice
choi
ces,
as w
ell.
Uni
t fou
r – 1
04
But w
hat i
s th
e le
ngth
of t
he le
g ad
jace
nt to
𝛼?
We
can
figur
e th
is ou
t bec
ause
this
is a
spec
ial r
ight
tria
ngle
, viz
., it
is a
30
°−
60
°−
90
° tr
iang
le. S
ince
our
hyp
oten
use
is 1
, the
sh
ort
leg
(not
mar
ked)
mus
t mea
sure
1 2. T
hen,
to
find
the
long
leg,
we
mul
tiply
1 2 b
y √
3.
Thus
the
long
leg,
𝑎, m
ust b
e √3 2. S
ince
𝑎=
√3 2, w
e co
nclu
de th
at
cos𝛼
=
√3 2 1
=√
3 2.
We
have
one
mor
e Tr
igon
omet
ric f
unct
ion
to u
ncov
er. W
e’ll
defin
e it
here
,but
in t
he
exer
cise
s yo
u w
ill d
eriv
exiii i
t you
rsel
f.
The
Tang
ent f
unct
ion
Th
e Ta
ngen
t fun
ctio
n ac
cept
s as
an
inpu
t an
acut
e an
gle,
and
then
retu
rns
the
ratio
of
the
side
oppo
site
to th
e an
gle
to th
e si
de a
djac
ent t
o th
e an
gle.
Sym
bolic
ally
, ta
n𝛼
=𝑎 𝑏
.
We
coul
d al
so s
tate
that
tan𝛽
=𝑏 𝑎
.
The
Tang
ent f
unct
ion
appe
ars
a bi
t diff
eren
t fro
m th
e Si
ne a
nd C
osin
e fu
nctio
n, a
nd, t
o be
sur
e, th
ere
are
som
e ke
y di
ffere
nces
. But
you
’ll a
lso d
iscov
er in
the
exer
cise
s th
at it
is
clos
ely
rela
ted
to t
he S
ine
and
Cosin
e fu
nctio
ns! A
dditi
onal
ly –
as
you’
ll so
on s
ee –
we
wor
kw
ith it
in th
e sa
me
man
ner.
Exam
ple
3a
Writ
e th
e Si
ne, C
osin
e, a
nd T
ange
nt ra
tios
usin
g 𝛼
of t
he tr
iang
le s
how
n in
Fig
ure
37.
xi
ii Wha
t thi
s m
eans
is th
at y
ou’ll
take
som
ethi
ng y
ou a
lread
y kn
ow, w
ork
with
it, a
nd c
ome
out w
ith th
e fu
nctio
n w
e’re
abo
ut to
def
ine!
Obv
ious
ly th
is is
the
bett
er w
ay to
do
it, b
ut w
e do
n’t w
ant t
o sp
oil y
our
fun.
§2
The
Sin
e, C
osin
e, a
nd T
ange
nt fu
nctio
ns –
105
All w
e ne
ed to
do
is s
ubst
itute
. The
onl
y pr
oble
m w
e ha
ve is
that
we
don’
t hav
e a
leng
th
for
the
hypo
tenu
se. N
o m
atte
r –
this
is e
asily
foun
d. W
e us
e Py
thag
oras
’ The
orem
and
fin
d th
at th
e hy
pote
nuse
mus
t hav
e a
leng
th o
f 8.54
. With
this
in m
ind,
we
have
sin𝛼
=8
8.54
,
cos𝛼
=3 8.4
,
and
tan𝛼
=8 3
.
Exam
ple
3b
Eval
uate
tan4
5°.
We
follo
w t
he s
ame
proc
edur
e as
Exa
mpl
es 2
c, 1
c, a
nd 1
d. W
e cr
eate
any
siz
e rig
ht
tria
ngle
with
a 4
5° a
ngle
, as
in F
igur
e 38
.
Now
all
we
need
to d
o is
writ
e ou
r rat
io. W
e ha
ve
tan4
5°
=1 1
=1
.
Thus
we
conc
lude
that
tan4
5° i
s 1
.
Figu
re 3
7
Figu
re 3
8
Reca
ll th
at a
ny 4
5°−4
5°−
90
° tria
ngle
will
also
be
isosc
eles
. Bot
h le
gs, t
here
fore
, are
also
con
grue
nt.
Uni
t fou
r – 1
06
Usu
ally
, eva
luat
ing
a Tr
ig f
unct
ion
give
n an
y in
put
is v
ery
diffi
cult
to d
o. O
f co
urse
, in
toda
y’s
wor
ld, w
e ha
ve r
eady
acc
ess
to c
alcu
lato
rs, w
hich
are
ver
y go
od a
t th
e te
diou
s pr
oces
s of
app
roxi
mat
ing,
so
it isn
’t to
o di
fficu
lt to
eva
luat
e si
n2
°, f
or e
xam
ple.
You
’ll
need
to h
ave
a ca
lcul
ator
on
hand
.
Exam
ple
4a
Det
erm
ine
the
leng
th o
f 𝑎 g
iven
Fig
ure
39.
As w
e di
d ex
tens
ivel
y in
Geo
met
ry,
we
shou
ld f
irst
esta
blish
som
e re
latio
nshi
p. F
or
exam
ple,
we
know
(fro
m P
ytha
gora
s) t
hat 𝑎
2+𝑏
2=𝑐
2. I
n th
is ca
se, h
owev
er, w
e do
n’t
know
𝑎 o
r 𝑏, a
nd s
o Py
thag
oras
’ The
orem
yie
lds
us n
o he
lp. S
ince
we
have
bee
n gi
ven
an
angl
e m
easu
re a
nd o
ne si
de le
ngth
, it s
tand
s to
reas
on th
at w
e sh
ould
use
a T
rig fu
nctio
n.
We
have
thre
e di
ffere
nt c
hoic
es (S
ine,
Cos
ine,
and
Tan
gent
), so
whi
ch o
ne sh
ould
we
use?
In
thi
s ex
ampl
e, w
e w
ill tr
y al
l of t
hem
and
sho
w y
ou t
hat
ther
e is
rea
lly o
nly
one
good
ch
oice
.xiv
Firs
t, le
t us
try
the
Sin
e fu
nctio
n. G
iven
tha
t si
n𝛼
=opposite
hypotenuse
, we
have
(af
ter
subs
titut
ion)
sin2
0°
=𝑎 45
.
If w
e tr
y th
e Co
sine
func
tion,
we
have
cos2
0°
=ad
jace
nt
hyp
ote
nu
se=
? 45
,
whe
re th
e qu
estio
n m
ark
repr
esen
ts th
e un
know
n le
ngth
of t
he a
djac
ent.
If w
e tr
y th
e Ta
ngen
t fun
ctio
n, w
e w
ould
hav
e
tan2
0°
=o
pp
osi
te
adja
cen
t=𝑎 ?
,
whe
re, a
gain
, the
que
stio
n m
ark
repr
esen
ts th
e sa
me
unkn
own
leng
th o
f the
adj
acen
t.
xi
v Thi
s w
on’t
alw
ays
be th
e ca
se. S
omet
imes
ther
e w
ill b
e tw
o or
thre
e go
od c
hoic
es.
Figu
re 3
9
§2
The
Sin
e, C
osin
e, a
nd T
ange
nt fu
nctio
ns –
107
Base
d on
wha
t w
e ju
st s
aw, t
he S
ine
is t
he fu
nctio
n w
e w
ill c
hoos
e. T
his
is b
ecau
se w
e ha
ve a
n eq
uatio
n,
sin2
0°
=𝑎 45
,
whi
ch c
an b
e so
lved
for t
he m
issin
g va
riabl
e 𝑎
. If w
e us
ed th
e Co
sine
func
tion,
we
coul
d fin
d th
e le
ngth
of t
he a
djac
ent,
but t
hat w
ould
not
get
us
the
leng
th o
f 𝑎.xv
The
Tan
gent
fu
nctio
n is
even
wor
se; w
e w
ould
hav
e tw
o va
riabl
es in
our
equ
atio
n, w
hich
wou
ld p
ut u
s in
an
unte
nabl
e po
sitio
n.
To s
olve
this
equa
tion,
we
shou
ld e
valu
ate
the
left
side,
and
then
get
the
varia
ble
all b
y its
elf.
Usin
g a
calc
ulat
or, w
e fin
d th
at s
in2
0°≈
0.342
. Hen
ce
0.342
=𝑎 45
15
.39≈𝑎
.
The
key
to th
ese
prob
lem
s is
to id
entif
y th
e co
rrec
t rel
atio
nshi
p. T
hen
it’s
just
a m
atte
r of
subs
titut
ing
and
solv
ing,
bot
h of
whi
ch a
re b
asic
Alg
ebra
ski
lls.
Exam
ple
4b
Det
erm
ine
the
leng
th o
f 𝑏 g
iven
Fig
ure
40.
We
face
a s
imila
r pro
blem
. Her
e th
e be
st re
latio
nshi
p to
use
is T
ange
nt, s
ince
we
can
then
fo
rm th
e eq
uatio
n
tan
50
°=
7.8 𝑏
.
Then
we
just
sol
ve th
e eq
uatio
n w
henc
e
xv
That
bei
ng s
aid,
we
coul
d fin
d th
e le
ngth
of 𝑎
afte
r we
find
the
leng
th o
f the
adj
acen
t. It’
s no
t an
idea
l pr
oces
s th
ough
, as
we
have
to g
o th
roug
h th
e en
tire
proc
edur
e ag
ain,
or,
alte
rnat
ivel
y, u
se P
ytha
gora
s’ Th
eore
m. E
ither
way
, it’s
muc
h m
ore
effic
ient
to u
se th
e Si
ne fu
nctio
n.
Figu
re 4
0
Uni
t fou
r – 1
08
𝑏≈
6.54
.
Exam
ple
4c
Com
plet
e th
e rig
ht tr
iang
le s
how
n in
Fig
ure
41.
Reca
ll th
at w
hen
we
are
aske
d to
“com
plet
e” a
righ
t tria
ngle
, it j
ust m
eans
that
we
shou
ld
find
out t
he le
ngth
of e
very
sid
e an
d th
e m
easu
re o
f eve
ry a
ngle
.xvi
We
have
som
e fre
edom
in
how
we
appr
oach
thi
s pr
oble
m,
but
ther
e ar
e al
so s
ome
limita
tions
.xvii L
et u
s fir
st fi
nd 𝑎
. To
do th
is, w
e m
ust a
sk th
at e
ver-
popu
lar q
uest
ion:
Wha
t is
the
rela
tions
hip?
In th
is ca
se, w
e co
uldxv
iii u
se th
e Si
ne fu
nctio
n to
get
sin
58
°=
𝑎 12
0.8
5≈
𝑎 12
𝑎≈
10
.18
.
Now
that
we
have
𝑎, w
e ca
n ea
sily
find 𝑏
by
usin
g Py
thag
oras
’ The
orem
. We
can
also
find
𝛽
eas
ily, s
ince
the
sum
of t
he th
ree
angl
es m
ust b
e 1
80
°. H
ence
𝑏≈
6.3
5,𝛽
=32
°.
Befo
re y
ou b
egin
pra
ctic
ing
thes
e co
ncep
ts, t
here
are
thr
ee o
ther
Trig
fun
ctio
ns w
hich
ex
ist. T
hese
are
rare
ly u
sed
by th
emse
lves
, but
they
will
be
of s
ome
impo
rtan
ce la
ter.
We
men
tion
them
her
e fo
r the
sak
e of
com
plet
ion.
xv
i Tha
t mak
es a
tota
l of s
ix th
ings
we’
re lo
okin
g fo
r, al
thou
gh w
e m
ust b
e gi
ven
at le
ast t
hree
of t
hem
to
dete
rmin
e th
e ot
her t
hree
. xv
ii For
exa
mpl
e, c
ould
we
find
any
of th
e si
des
usin
g Py
thag
oras
’ The
orem
? xv
iii W
e sa
y “c
ould
” bec
ause
we
coul
d al
so fi
nd th
e le
ngth
of 𝑏
by
usin
g th
e Co
sine
func
tion.
Cou
ld w
e us
e th
e Ta
ngen
t fun
ctio
n he
re?
Why
or w
hy n
ot?
Figu
re 4
1
§2
The
Sin
e, C
osin
e, a
nd T
ange
nt fu
nctio
ns –
109
The
reci
proc
al T
rigon
omet
ric fu
nctio
ns
Th
e Co
seca
nt fu
nctio
n:
csc𝛼
=𝑐 𝑎
The
Seca
nt fu
nctio
n:
sec𝛼
=𝑐 𝑏
The
Cota
ngen
t fun
ctio
n:
cot𝛼
=𝑏 𝑎
The
nam
e “r
ecip
roca
l Trig
func
tions
” is
very
apr
opos
. See
if y
ou c
an fi
nd o
ut w
hy.
§𝟐 E
xerc
ises
1.)
Writ
e th
e ra
tios
of th
e Si
ne, C
osin
e, a
nd T
ange
nt fu
nctio
n (fo
r bot
h 𝛼
and
𝛽) g
iven
th
e fo
llow
ing
tria
ngle
s.
(A)
(B)
(C)
(D)
2.)
Our
nex
t goa
l is
to c
reat
e a
tabl
e fo
r the
Sin
e, C
osin
e, a
nd T
ange
nt fu
nctio
ns. F
irst,
how
ever
, we’
ll on
ly fo
cus
on t
hose
rea
lly n
ice
angl
es w
hich
pro
duce
spe
cial
rig
ht
tria
ngle
s. (A
) W
hat
are
the
“rea
lly n
ice
angl
es”
of w
hich
we
spea
k? (
Hin
t: T
hink
of
the
angl
es s
een
in a
ll of
our
spe
cial
righ
t tria
ngle
s.)
Uni
t fou
r – 1
10
(B)
Dra
w a
ny tw
o tr
iang
les
with
the
info
rmat
ion
from
(A).
Mak
e su
re y
ou la
bel
the
sides
and
ang
les.
(C)
Now
cre
ate
a ta
ble
of v
alue
s fo
r th
e Si
ne, C
osin
e, a
nd T
ange
nt f
unct
ions
. U
se (a
nd e
xpan
d) th
e in
com
plet
e ta
ble
belo
w a
s a
mod
el.
𝜶
𝐬𝐢𝐧𝜶
3
0°
1 2
45
° √2 2
3.)
We
have
not
yet
def
ined
sin
0°
and
cos
0°,
so l
et’s
do t
hat
now
. Co
nsid
er t
he
tria
ngle
bel
ow, w
here
𝛼 is
a v
ery
smal
l ang
le.
Not
e th
at, i
n th
e ab
ove
case
, sin𝛼
will
be
rela
tivel
y sm
all (
whi
ch y
ou s
houl
d ve
rify)
.
(A)
Wha
t will
hap
pen
to th
e le
ngth
of t
he s
ide
oppo
site
of 𝛼
as
the
angl
e of
𝛼
decr
ease
s?
(B)
Fina
lly, i
mag
ine
that
𝛼=
0. W
hat
will
the
leng
th o
f the
sid
e op
posi
te t
o 𝛼
be
? (C
) G
iven
you
r ans
wer
for (
B), w
hat m
ust s
in0
° be
equa
l to?
(D
) N
ow l
et’s
cons
ider
the
Cos
ine
func
tion.
Alre
ady,
the
len
gth
of t
he s
ide
adja
cent
to 𝛼
and
the
leng
th o
f the
hyp
oten
use
are
very
clo
se. W
hat i
s co
s𝛼
gi
ven
the
pict
ure?
(E
) As
𝛼 g
ets c
lose
to z
ero,
wha
t will
hap
pen
to th
e le
ngth
of t
he h
ypot
enus
e in
re
latio
n to
the
leng
th o
f the
adj
acen
t sid
e? (H
int:
Try
dra
win
g sm
alle
r and
sm
alle
r ang
les
for 𝛼
) (F
) G
iven
you
r ans
wer
for (
E), w
hat m
ust c
os
0° b
e eq
ual t
o?
(G)
Verif
y yo
ur re
sults
for (
C) a
nd (F
) with
a c
alcu
lato
r. 4.
) N
ow le
t’s d
efin
e th
e re
sults
of s
in9
0° a
nd c
os
90
°. (A
) W
e sh
ould
try
to
use
a sim
ilar
appr
oach
as
the
prev
ious
pro
blem
. Dra
w a
rig
ht tr
iang
le w
here
𝛼 is
clo
se to
90
°, bu
t not
qui
te th
at la
rge.
(B
) Ba
sed
on y
our p
ictu
re, w
hat w
ill s
in9
0° a
nd c
os
90
° be?
(C
) Ve
rify
your
resu
lts w
ith a
cal
cula
tor.
5.)
Crea
te a
tabl
e of
val
ues
for a
ll th
ree
Trig
func
tions
from
0° t
o 9
0°.
Incr
emen
t eac
h ro
w b
y 5
° (s
o yo
ur f
irst
num
ber
is 0
°, th
en 5
°, th
en1
0°…
). Yo
u sh
ould
use
a
calc
ulat
or fo
r thi
s pr
oble
m.
§2
The
Sin
e, C
osin
e, a
nd T
ange
nt fu
nctio
ns –
111
6.)
Com
plet
e th
e fo
llow
ing
right
tria
ngle
s.
(A)
(B)
(C)
(D)
(E)
(F)
7.
) Th
e pr
evio
us c
once
pts
can
also
be
appl
ied
to q
uasi-
real
sce
nario
s. (A
) Be
nny
lean
s a
ladd
er a
gain
st a
24
foot
wal
l, so
tha
t th
e to
p of
the
ladd
er
touc
hes
the
top
of t
he w
all p
erfe
ctly
. The
ang
le t
hat
is cr
eate
d w
ith t
he
grou
nd a
nd th
e la
dder
is 7
0°.
Det
erm
ine
the
leng
th o
f the
ladd
er.
(B)
Build
ing
A is
sho
rter
than
Bui
ldin
g B.
To
dete
rmin
e ho
w m
uch
shor
ter,
the
may
or la
ys a
tape
mea
sure
from
Bui
ldin
g A
to B
uild
ing
B an
d se
es th
at th
e sla
nt h
eigh
t is 4
5 m
. The
n he
also
figu
res
out
that
the
ang
le b
etw
een
the
tape
mea
sure
and
Bui
ldin
g B
is 62
°. H
ow m
uch
shor
ter
is Bu
ildin
g A
than
Bu
ildin
g B?
(C
) A
wat
er sl
ide
is h
ouse
d in
side
a to
wer
that
is 5
0 ft
tall.
The
wat
er sl
ide
trav
els
in a
str
aigh
t lin
e do
wn
tow
ard
the
grou
nd, w
here
it h
its th
e gr
ound
at a
24
° an
gle.
How
muc
h sp
ace
is n
eede
d on
the
grou
nd fo
r thi
s sli
de?
(D)
A tr
ee’s
shad
ow is
10
ft lo
ng. T
he a
ngle
tha
t th
e su
n cr
eate
s w
ith t
he fl
at
grou
nd is
80
°. D
eter
min
e th
e he
ight
of t
he tr
ee.
8.)
Crea
te y
our o
wn
quas
i-rea
l situ
atio
n, s
imila
r to
one
of th
e pr
evio
us p
robl
ems,
that
re
quire
s Tr
igon
omet
ry to
sol
ve.
9.)
Look
at t
he ta
ble
of v
alue
s yo
u cr
eate
d fo
r the
Sin
e fu
nctio
n.
(A)
Wha
t is
the
max
imum
val
ue o
f sin𝛼
?
Uni
t fou
r – 1
12
(B)
Wha
t is
the
min
imum
val
ue o
f sin𝛼
? (C
) Ar
e th
ose
max
imum
and
min
imum
val
ues
true
for
any
val
ue o
f 𝛼
? Tr
y si
n5
00
° or s
in( −
10
0°)
to v
erify
. (D
) D
escr
ibe
any
patt
erns
you
see
with
the
Sine
func
tion.
10
.) Lo
ok a
t the
tabl
e of
val
ues
you
crea
ted
for t
he C
osin
e fu
nctio
n.
(A)
Wha
t is
the
max
imum
val
ue o
f co
s𝛼
? (B
) W
hat i
s th
e m
inim
um v
alue
of c
os𝛼
? (C
) Ar
e th
ose
max
imum
and
min
imum
val
ues
true
for a
ny v
alue
of 𝛼
? (D
) D
escr
ibe
any
patt
erns
you
see
with
the
Cosin
e fu
nctio
n.
11.)
Look
at t
he ta
ble
of v
alue
s yo
u cr
eate
d fo
r the
Tan
gent
func
tion.
(A
) W
hat i
s th
e m
axim
um v
alue
of t
an𝛼
? (B
) W
hat i
s th
e m
inim
um v
alue
of t
an𝛼
? (C
) Ar
e th
ose
max
imum
and
min
imum
val
ues
true
for a
ny v
alue
of 𝛼
? (D
) D
escr
ibe
any
patt
erns
you
see
with
the
Tang
ent f
unct
ion.
12
.) Im
agin
e fo
r a m
omen
t tha
t you
mea
sure
a tr
iang
le in
yar
ds, a
nd y
ou fi
nd th
at th
e tw
o le
gs m
easu
re 3
and 4
yar
ds, w
hile
the
hypo
tenu
se m
easu
res
5 y
ards
. You
sen
d th
is in
form
atio
n to
you
r frie
nd w
ho li
ves
in a
Eur
opea
n co
untr
y th
at n
orm
ally
use
s m
eter
s in
stea
d.
(A)
Calc
ulat
e si
n𝛼
usin
g th
e in
itial
cor
rect
mea
sure
men
t tha
t use
d ya
rds.
(B)
Now
con
vert
yar
ds in
to m
eter
s (ro
und
to th
e ne
ares
t hun
dred
th).
Calc
ulat
e si
n𝛼
in m
eter
s. (C
) As
sum
e th
at w
hen
you
sent
thi
s tr
iang
le t
o yo
ur fr
iend
, you
did
not
labe
l yo
ur m
easu
rem
ents
. So
your
frie
nd a
ssum
es y
ou m
ean
3 m
eter
s, 4
met
ers,
and
so o
n. C
alcu
late
sin𝛼
with
this
inco
rrec
t inf
orm
atio
n.
(D)
Was
you
r an
swer
any
diff
eren
t in
(A)
, (B)
, or
(C)?
Wha
t do
es t
hat
tell
you
abou
t Sin
e an
d Co
sine
ratio
s (v
iz.,
do u
nits
mat
ter)?
13
.) O
ne p
opul
ar m
nem
onic
dev
ice
for
rem
embe
ring
the
diffe
rent
Trig
rat
ios
is S
OH
CA
H T
OA.
Exp
lain
wha
t thi
s m
eans
and
how
it h
elps
you
to re
mem
ber h
ow to
set
up
you
r rat
ios.
(If y
ou d
on’t
know
it, l
ook
it up
) 14
.) Le
t us
now
der
ive
the
Tang
ent f
unct
ion.
To
acco
mpl
ish th
is, w
e on
ly n
eed
the
Sine
an
d Co
sine
func
tion.
(A
) W
hen
deriv
ing
thin
gs, i
t is
ofte
n us
eful
to r
ewrit
e th
em in
thei
r mos
t bas
ic
term
s. Be
fore
we
can
do th
at, h
owev
er, l
et u
s dr
aw a
righ
t tria
ngle
. Cho
ose
any
non-
right
ang
le a
nd c
all i
t 𝛼
. The
n la
bel t
he s
ides
(acc
ordi
ng t
o yo
ur
choi
ce o
f 𝛼) a
s “o
ppos
ite”,
“adj
acen
t”, a
nd “h
ypot
enus
e”.
(B)
Usin
g yo
ur p
ictu
re, w
rite
out t
he S
ine
and
Cosin
e ra
tios.
(C)
Is th
ere
any
way
we
can
take
our
Sin
e an
d Co
sine
ratio
and
turn
them
into
ou
r Ta
ngen
t ra
tio?
Reca
ll th
at T
ange
nt is
opposite
adjacent. T
ry a
ddin
g, s
ubtr
actin
g,
mul
tiply
ing,
or d
ivid
ing
them
.
§2
The
Sin
e, C
osin
e, a
nd T
ange
nt fu
nctio
ns –
113
(D)
Why
is it
bet
ter t
o de
rive
the
tang
ent f
unct
ion
than
to d
efin
e it?
Put
ano
ther
w
ay, w
hat i
ssue
s w
ould
you
run
into
if y
ou ju
st d
efin
ed e
very
thin
g in
mat
h?
§3
Re
latio
nshi
ps b
etw
een
the
Trig
func
tions
In o
ur f
inal
sec
tion,
we’
ll lo
ok a
t ho
w t
he t
hree
mai
n Tr
ig f
unct
ions
are
rel
ated
to
one
anot
her.
We
alre
ady
know
a f
ew r
elat
ions
hips
, vi
z. th
at t
he C
osin
e fu
nctio
n is
the
com
plem
ent t
o th
e si
ne fu
nctio
n. S
o it
is fa
ir to
bel
ieve
that
ther
e ar
e ot
her r
elat
ions
hips
th
at e
xist
, and
in th
is se
ctio
n, w
e w
ill te
ase
them
out
. We
will
spe
nd o
ur ti
me
deriv
ing
and
form
aliz
ing
man
y of
the
basic
and
fund
amen
tal i
dent
itie
s in
this
sect
ion.
An
iden
tity
is a
di
ffere
nt w
ay to
writ
e an
equ
ival
ent s
tate
men
t. Fo
r exa
mpl
e, 2
+2
=4
is a
n id
entit
y; 2
+2
is
sim
ply
a di
ffere
nt n
ame
for 4
.
Befo
re w
e em
bark
on
lear
ning
som
e id
entit
ies,
let u
s fir
st lo
ok a
t a m
ore
basic
que
stio
n:
Doe
s kn
owin
g on
e Tr
ig r
atio
lead
us
to f
ind
any
of t
he o
ther
s? P
ut a
noth
er w
ay, i
f w
e kn
ow o
ne T
rig ra
tio, c
an w
e fin
d al
l of t
he o
ther
?
Exam
ple
1a
If si
n𝛼
=3 5, w
rite
the
othe
r six
Trig
ratio
s.
Let
us f
irst
draw
a p
ictu
re.
Beca
use
the
Sine
rat
io i
s th
e sid
e op
posi
te t
o 𝛼
to
the
hypo
tenu
se, i
t mak
es s
ense
to d
raw
Fig
ure
42.
This
draw
ing
mak
es o
ur li
fe m
uch
easie
r. N
ow w
hat w
e’ll
do is
det
erm
ine
the
othe
r rat
ios.
Star
ting
with
Cos
ine,
we
wou
ld h
ave co
s𝛼
=ad
jace
nt
5.
Figu
re 4
2
As b
efor
e, th
is isn
’t th
e on
ly tr
iang
le th
at y
ou c
ould
dra
w.
Uni
t fou
r – 1
14
But w
hat i
s the
leng
th o
f the
adj
acen
t sid
e? T
his c
an b
e ea
sily
figur
ed o
ut w
ith P
ytha
gora
s’ Th
eore
m, a
lthou
gh, i
n th
is ca
se, w
e re
cogn
ize
that
we
have
a P
ytha
gore
an T
riple
, and
thus
th
e ad
jace
nt s
ide
is 4
. With
this
in m
ind,
the
rest
of t
he T
rig ra
tios
are
elem
enta
ry. T
hey
are
cos𝛼
=4 5
,ta
n𝛼
=3 4
,cs
c𝛼
=5 3
,se
c𝛼
=5 4
,co
t𝛼
=4 3
.
Exam
ple
1b
If ta
n𝛽
=5 7, w
rite
the
othe
r six
Trig
ratio
s.
You
shou
ld, a
gain
, sta
rt w
ith a
pic
ture
.i We
draw
the
tria
ngle
sho
wn
in F
igur
e 43
.
Befo
re w
e w
rite
out a
ny o
f our
ratio
s, ho
w a
bout
we
find
the
leng
th o
f the
miss
ing
side,
w
hich
in
the
pres
ent
case
is
the
hypo
tenu
se.
Sinc
e w
e do
not
rec
ogni
ze t
his
as a
Py
thag
orea
n Tr
iple
, we
go a
head
and
use
Pyt
hago
ras’
Theo
rem
. We
find
that
52
+7
2=𝑐
2
𝑐=√
74
.ii
With
this
valu
e, w
e ca
n no
w w
rite
all s
ix ra
tios:
sin𝛼
=5
√74
,co
s𝛼
=7
√74
,ta
n𝛼
=5 7
,cs
c𝛼
=√
74
5,
sec𝛼
=√
74
7,
cot𝛼
=7 5
.
i S
ensi
ng a
pat
tern
yet
? ii Y
ou s
houl
d al
way
s le
ave
this
num
ber i
n ex
act f
orm
. Sim
plify
it if
you
can
.
Figu
re 4
3
§3
Rel
atio
nshi
ps b
etw
een
the
Trig
func
tions
– 1
15
Not
e th
at it
isn’
t ne
cess
ary
to r
atio
naliz
e th
e de
nom
inat
ors,
alth
ough
it m
ight
be
good
pr
actic
e fo
r yo
u to
do
so. A
lso n
ote
that
som
e te
xtbo
oks
and
stan
dard
ized
tes
ts w
ill
requ
ire y
ou to
do
this,
so
mak
e su
re y
ou k
now
how
to d
o th
is.
Exam
ple
1c
Giv
enth
at s
in𝛼
=𝑎
, fin
d al
l six
Trig
ratio
s.
This
see
ms
mor
e di
fficu
lt th
an t
he p
revi
ous
prob
lem
s. D
o no
t le
t fir
st a
ppea
ranc
es
intim
idat
e yo
u –
you
just
nee
d to
dra
w a
pic
ture
. The
onl
y iss
ue w
ith o
ur p
ictu
re is
tha
t w
e on
ly s
eem
to h
ave
the
leng
th o
f one
sid
e, 𝑎
. But
reca
ll th
at 𝑎
=𝑎 1
, so
we
can
draw
our
pict
ure
as s
een
in F
igur
e 44
.
So w
hat’s
the
len
gth
of t
head
jace
nt s
ide?
We
can
figur
e th
is o
ut u
sing
Pyt
hago
ras’
Theo
rem
, whi
ch, u
nsur
prisi
ngly
, is
exac
tly w
hat w
e di
d in
the
prev
ious
Exa
mpl
es. W
e le
t 𝑏
be th
e le
ngth
of t
he a
djac
ent s
ide,
and
set
up
our e
quat
ion
like
so:
𝑎2
+𝑏
2=
12
.
We
will
let y
ou fi
nish
this
Exam
ple
in th
e Ex
erci
ses.
We
have
see
n ho
w w
e ca
n re
late
one
Trig
rat
io t
o th
e ot
hers
. We
will
nex
t re
late
the
fu
nctio
ns th
emse
lves
.
Ago
od s
tart
ing
poin
t is
to b
egin
with
wha
t we
know
, and
bui
ld fr
om th
ere.
We
know
that
(1)
cos𝛼
=si
n𝛽
iff 𝛼
+𝛽
=9
0°.
This
is a
fine
rela
tions
hip,
but
let’s
mak
e th
is m
ore
usef
ul. O
ne is
sue
is th
at th
ere
are
two
varia
bles
, so
let’s
try
to s
impl
ify th
at s
tate
men
t.
Figu
re 4
4
Uni
t fou
r – 1
16
Exam
ple
2a
Rew
rite
(1) i
n te
rms
of 𝛼
.iii
We
are
give
n th
at c
os𝛼
=si
n𝛽
iff 𝛼
+𝛽
=9
0°.
We
can
thus
mak
e a
subs
titut
ion.
Sin
ce
𝛼+𝛽
=9
0°
then
we
know
that
𝛽=
90
°−𝛼
and
henc
e
cos𝛼
=si
n( 9
0°−𝛼) .
Exam
ple
2b
Giv
enco
s2
0°,
wha
t is
the
equi
vale
nt c
ofun
ctio
n?
To a
nsw
er t
his,
we
need
onl
y us
e th
e pr
evio
us r
esul
t. In
the
pre
sent
cas
e, 𝛼
=2
0°
and
henc
e
cos2
0°
=si
n( 9
0°−2
0°)
=si
n7
0°.
The
answ
er w
e’re
loo
king
for
is
sin
70
°. A
calc
ulat
or c
an q
uick
ly v
erify
tha
t in
deed
, co
s2
0°
=si
n7
0°.
Do
reco
gniz
e th
at th
e co
func
tion
of C
osin
e is
Sin
e, a
nd v
ice
vers
a. W
hich
Trig
func
tion
is
the
cofu
nctio
n of
Tan
gent
? O
r how
abo
ut C
osec
ant?
Is th
ere
an e
asy
way
to te
ll?
Exam
ple
2c
Writ
e al
l of t
he c
ofun
ctio
ns id
enti
ties
.
We
will
not
writ
e al
l of t
hem
, but
inst
ead,
will
writ
e on
e of
them
and
leav
e th
e re
st to
the
read
er.
sin𝛼
=co
s(9
0°−𝛼) .
Oth
er c
ofun
ctio
n id
entit
ies
exist
, and
you
will
hav
e to
writ
e th
em o
ut in
you
r exe
rcis
es.
The
cofu
nctio
n id
entit
ies
can
be u
sefu
l, bu
t m
ostly
the
y ju
st h
ighl
ight
the
rel
atio
nshi
p be
twee
n th
e Si
ne a
nd C
osin
e fu
nctio
n (a
nd th
e ot
her c
o-Tr
ig fu
nctio
ns).
iii In
oth
er w
ords
, the
re s
houl
d on
ly b
e on
e va
riabl
e, 𝛼
.
§3
Rel
atio
nshi
ps b
etw
een
the
Trig
func
tions
– 1
17
Let u
s no
w e
xplo
re w
hat h
appe
ns w
hen
we
squa
re th
e Si
ne o
r Co
sine
func
tions
. Firs
t of
all,
if w
e w
rite
sin𝛼
2 th
ere
coul
d be
a b
it of
con
fusi
on.iv
So,
if w
e m
ean ( s
in𝛼)(
sin𝛼) ,
we
will
writ
e
sin
2𝛼
.
If w
e m
ean
sin( 𝛼
2) ,
whe
re t
he a
ngle
𝛼 is
squ
ared
but
not
the
func
tion,
we’
ll w
rite
it as
si
n( 𝑎
2) .
Exam
ple
3
Eval
uate
sin
2𝛼
giv
en F
igur
e 45
.
Reca
ll th
at s
in2𝛼
=( s
in𝛼)(
sin𝛼) ,
and
sin
ce s
in𝛼
=3 5, w
e ha
ve, b
y su
bstit
utio
n,
(3 5)(3 5)
9 25
.
You’
ll w
ork
to e
xplo
re s
quar
ed T
rig fu
nctio
ns in
the
exer
cise
s.
Now
let’s
see
wha
t hap
pens
whe
n w
e ad
d tw
o sq
uare
d fu
nctio
ns.
Let’s
try
sin
24
5°
+co
s24
5°.
Sinc
e si
n4
5°
=√
2 2, w
e ca
n co
nclu
de th
at
sin
24
5°
=2 4
=1 2
.
This
is a
lso
true
of c
os24
5°,
since
co
s4
5°
=√
2 2. T
hus
we
have
sin
24
5°
+co
s24
5°
=1 2
+1 2
=1
.
This
is q
uite
nic
e. W
ill th
is al
way
s be
true
?
iv A
re w
e sq
uarin
g th
e an
gle 𝛼
or s
quar
ing
the
func
tion
sin𝛼
?
Figu
re 4
5
Uni
t fou
r – 1
18
Exam
ple
4
Eval
uate
sin
26
0°
+co
s26
0°.
From
the
prev
ious
sec
tion,
we
have
mem
oriz
ed t
hat
sin
60
°=
√3 2.v T
hus
sin
26
0°
=3 4. W
e
also
hav
e m
emor
ized
co
s6
0°
=1 2, a
nd th
us c
os2
60
°=
1 4. W
e th
usly
con
clud
e th
at
sin
26
0°
+co
s26
0°
=3 4
+1 4
=1
.
But
that
’s st
rang
e… W
hy h
ave
we
gott
en t
he s
ame
answ
er?
And
why
is t
hat
answ
er s
o pl
easa
nt?
Hav
e w
e st
umbl
ed u
pon
som
e re
mar
kabl
e tr
uth?
Pyth
agor
ean
iden
tity
For a
ny a
ngle
𝛼,
sin
2𝛼
+co
s2𝛼
=1
. W
e’ll
revi
sit th
is id
entit
y in
gra
phic
al fo
rmat
, but
that
will
be
afte
r we
reve
al th
e U
nit C
ircle
. It
is th
ere
that
we
will
pro
ve th
is id
entit
y, a
s it i
s ver
y ea
sy to
do
whe
n w
e vi
ew it
gra
phic
ally
.
It w
ould
be
safe
to
ask
if th
is is
the
only
suc
h re
latio
nshi
p be
twee
n co
func
tions
. For
ex
ampl
e, is
tan
2𝛼
+co
t2𝛼
also
equ
al to
one
? Yo
u w
ill e
xplo
re th
at in
the
exer
cise
s.
In §2
, we
brie
fly i
ntro
duce
d th
e Re
cipr
ocal
Trig
fun
ctio
ns, s
uch
as S
ecan
t. Th
ese
are
defin
ed a
s th
e re
cipr
ocal
s of
thei
r res
pect
ive
Trig
func
tion.
In c
ase,
for e
xam
ple,
we
wan
t to
fin
d th
e ra
tio o
f th
e hy
pote
nuse
to
the
oppo
site
, we
wou
ld t
hen
use
the
Cose
cant
fu
nctio
n. A
nd s
ince
csc𝛼
=h
ypo
ten
use
op
po
site
whi
le
sin𝛼
=o
pp
osi
te
hyp
ote
nu
se,
we
reco
gniz
e th
at th
ese
are
reci
proc
als
(thus
the
nam
e).
Reca
ll th
at o
ne w
ay w
e ca
n w
rite
a re
cipr
ocal
is t
o pu
t it
unde
r on
e. L
et’s
put
that
into
sy
mbo
ls. If
𝑎 is
som
e nu
mbe
r, an
d w
e w
ant t
o fin
d its
reci
proc
al, w
e ca
n ju
st e
valu
ate
(or
v A
nd if
you
hav
en’t
mem
oriz
ed it
, tha
t’s o
k to
o. Y
ou w
ill th
en n
eed
to c
onst
ruct
a 3
0°−
60
°−
90
° tria
ngle
an
d w
rite
out t
he ra
tio, t
hen
sim
plify
wha
t you
hav
e.
§3
Rel
atio
nshi
ps b
etw
een
the
Trig
func
tions
– 1
19
leav
e, if
we
pref
er) 1 𝑎
. So
if w
e w
ant t
o fin
d th
e re
cipr
ocal
of t
he n
umbe
r 3 4, w
e ju
st n
eed
to
eval
uate
1 3 4
, whi
ch is
4 3, a
s yo
u sh
ould
ver
ify.
With
this
in m
ind
we
can
now
def
ine
the
Reci
proc
al id
entit
ies.
Reci
proc
al id
entit
ies
sin𝛼
=1
csc𝛼
,cs
c𝛼
=1
sin𝛼
Of c
ours
e, t
here
are
oth
er r
ecip
roca
l ide
ntiti
es, b
ut y
ou w
ill w
rite
the
rem
aind
er in
you
r ex
erci
ses.
Let u
s no
w p
ut th
ese
iden
titie
s to
use
.
Exam
ple
5a
Sim
plify
sin𝛼
csc𝛼
+cos𝛼
sec𝛼
.
The
key
to th
is p
robl
em is
to re
writ
e th
is e
xpre
ssio
n in
to s
omet
hing
eas
ier.
So w
hat w
e’ll
wan
t to
do
is id
entif
y so
me
aspe
ct o
f the
exp
ress
ion
that
can
be
rew
ritte
n. In
thi
s ca
se,
let’s
try
rew
ritin
g cs
c𝛼
and
see
wha
t we
get.
Sinc
e
sin𝛼
csc𝛼
=si
n𝛼∙
1
csc𝛼
and
1
csc𝛼
=si
n𝛼
, we
can
say
that
sin𝛼
csc𝛼
=si
n2𝛼
.
This
sam
e pr
oced
ure
will
reve
al th
at co
s𝛼
sec𝛼
=co
s2𝛼
.
So w
e ca
n re
writ
e ea
ch a
dden
d in
our
orig
inal
pro
blem
and
we
end
up w
ith
sin
2𝛼
+co
s2𝛼
,
whi
ch is
n’t t
oo b
ad. B
ut w
e ca
n re
writ
e th
at e
xpre
ssio
n as
som
ethi
ng e
ven
mor
e sim
ple!
Si
nce
that
is a
Pyt
hago
rean
iden
tity
and
is e
qual
to o
ne, o
ur fi
nal r
esul
t is
simpl
y
1.
How
did
we
know
to c
hang
e cs
c𝛼
and
sec𝛼
? N
othi
ng b
ut in
tuiti
on –
in o
ther
wor
ds, w
hen
we
wor
k w
ith t
hese
sor
ts o
f pr
oble
ms,
ther
e is
no
pres
crib
ed m
etho
d to
sim
plify
the
ex
pres
sion.
Exp
erie
nce
is a
big
hel
p, b
ut d
o no
t un
dere
stim
ate
plan
ning
and
pat
ienc
e,
Uni
t fou
r – 1
20
eith
er. A
nd o
f cou
rse,
you
mus
t be
wel
l-ver
sed
in th
e va
rious
iden
titie
s th
at w
e’ve
lear
ned
so fa
r. Yo
u’ll
find
that
if y
ou d
on’t
know
the
iden
titie
s ver
y w
ell t
hat s
impl
ifyin
gex
pres
sions
in
this
man
ner w
ill b
e ve
ry d
iffic
ult.
No
mat
ter h
ow g
ood
you
beco
me
at th
ese
iden
titie
s, ho
wev
er, b
e pr
epar
ed t
o sp
end
som
e tim
e w
ith t
hem
. Eve
n ve
ry g
ood
mat
hem
atic
ians
so
met
imes
stru
ggle
with
thes
e, so
don
’t fe
el in
com
pete
nt if
you
don
’t ge
t the
ans
wer
righ
t aw
ay.
One
rule
of t
hum
b w
hich
we’
ll re
itera
te to
you
: Yo
u w
ill m
ost l
ikel
y w
ant t
o co
nver
t any
Tr
ig fu
nctio
ns in
to S
ine
or C
osin
e, if
pos
sible
. We
have
man
y id
entit
ies
whi
ch w
ork
with
Si
ne a
nd C
osin
e, b
ut o
nly
a fe
w t
hat
wor
k w
ith t
he r
ecip
roca
l Trig
fun
ctio
ns. T
his
isn’t
alw
ays
the
case
, but
it’s
usua
lly th
e be
st p
lace
to s
tart
.
Exam
ple
5b
Sim
plify
sin𝛼
cos𝛼.
You
shou
ld h
ave
alre
ady
foun
d th
is ou
t in
the
Exer
cise
s in
the
prev
ious
sec
tion,
but
this
one
is so
impo
rtan
t tha
t goi
ng o
ver i
t a s
econ
d tim
e w
ill b
e he
lpfu
l to
you.
Not
e th
at th
e pr
oced
ure
we
use
here
will
ver
y ra
rely
be
used
by
you
in y
our
Exer
cise
s. Bu
t ag
ain,
the
re
sult
is s
o im
port
ant w
e fe
el it
nec
essa
ry to
incl
ude.
One
way
we
can
rew
rite
sin𝛼
and
co
s𝛼
is in
ter
ms
of t
heir
ratio
s. Bu
t ra
tios
requ
ire a
tr
iang
le, r
ight
? An
d w
e do
n’t
have
one
, so
wha
t sh
all w
e do
? W
ell,
how
abo
ut w
e m
ake
one?
Con
side
r Fig
ure
46, w
hich
will
allo
w u
s to
find
the
ratio
s of
our
two
Trig
func
tions
.
With
this
in m
ind,
we
can
now
sub
stitu
te, s
ince
sin𝛼
=𝑎 𝑐 a
nd c
os𝛼
=𝑏 𝑐, w
e ha
ve
sin𝛼
cos𝛼
=
𝑎 𝑐 𝑏 𝑐
.
Figu
re 4
6
This
isn’t
the
only
way
you
can
dra
w th
is tr
iang
le, e
ither
. The
key
is ju
st to
dra
w o
ne.
§3
Rel
atio
nshi
ps b
etw
een
the
Trig
func
tions
– 1
21
Reca
ll th
at o
ne c
an d
ivid
e tw
o fra
ctio
ns b
y m
ultip
lyin
g by
the
reci
proc
al. T
hus
we
have
𝑎 𝑐∙𝑐 𝑏
.
But 𝑐 𝑐
is o
ne, a
nd th
eref
ore
the 𝑐
valu
es c
ance
l out
. Thi
s le
aves
us
with
𝑎 𝑏.
This
is a
ratio
! And
in fa
ct, it
is n
o m
ore
than
the
Tang
ent f
unct
ion’
s rat
io. T
hus w
e co
nclu
de
that
sin𝛼
cos𝛼
=ta
n𝛼
.
You
will
use
this
iden
tity
man
y, m
any
times
, and
so
we
form
aliz
e it
belo
w.
Quo
tient
iden
tity
tan𝛼
=si
n𝛼
cos𝛼
Exam
ple
5c
Sim
plify
co
s𝛼∙t
an𝛼
.
Follo
win
g ou
r rul
e of
thum
b, le
t’s c
onve
rt e
very
thin
g in
to S
ines
and
Cos
ines
. To
do th
is,
we
sim
ply
rena
me
tan𝛼
usi
ng o
ur id
entit
y ab
ove.
We
then
hav
e
cos𝛼∙
sin𝛼
cos𝛼
whe
nce
we
see
that
the
Cosin
es c
ance
l. Th
is s
impl
y le
aves
us
with
sin𝛼
.
This
proc
ess
of s
impl
ifyin
g ex
pres
sion
s w
ill b
e fo
rmal
ized
in U
nit s
ix. U
ntil
then
, the
goa
l is
to in
trod
uce
you
to th
is ki
nd o
f thi
nkin
g.
§𝟑 E
xerc
ises
1.)
Use
the
give
n Tr
ig ra
tio to
writ
e ou
t all
six
Trig
ratio
s in
eac
h pr
oble
m.
(A)
sin𝛼
=1
2
13
(B)
tan𝛼
=4 3
(C)
cos𝛼
=7 25
(D)
sin𝛼
=1 4
(E)
cos𝛼
=10
13
(F)
tan𝛼
=5 6
(G)
csc𝛼
=7 2
(H)
sec𝛼
=10 7
Uni
t fou
r – 1
22
2.)
Let u
s co
mpl
ete
Exam
ple
1c. I
f sin𝛼
=𝑎
, wha
t are
the
six
Trig
ratio
s?
3.)
Now
let t
an𝛼
=𝑑
. (A
) W
hat a
re th
e six
Trig
ratio
s gi
ven
this?
(B
) Co
mpa
re th
is an
swer
to th
e pr
evio
us.
4.)
In E
xam
ple
1c, w
e le
t th
e le
ngth
of t
he h
ypot
enus
e of
the
tria
ngle
to
equa
l 1. I
s th
at O
K? W
hy d
on’t
we
let
the
hypo
tenu
se e
qual
𝑐,
to a
llow
for
any
and
all
poss
ibili
ties?
Now
sup
pose
sin𝛼
=𝑥 𝑐, a
nd th
e le
ngth
s of
you
r tria
ngle
are
𝑎,𝑏
, and
𝑐,
with
the
hypo
tenu
se e
qual
ing 𝑐.
(A
) W
hat a
re th
e six
Trig
ratio
s?
(B)
Com
pare
this
with
the
prev
ious
two
resu
lts.
5.)
Giv
en th
e Tr
ig fu
nctio
ns a
nd a
ngle
mea
sure
, writ
e th
e eq
uiva
lent
cof
unct
ion.
(A
) si
n3
0°
(B)
cos
10
° (C
) co
t7
° (D
) se
c64
° (E
) co
s3
1°
(F)
tan
14
° (G
) cs
c4
7°
(H)
sin2
5°
(I)
tan( 𝛽
+𝛾)
(J)
sin𝛽
6.
) W
rite
out a
ll of
the
cofu
nctio
n id
entit
ies,
incl
udin
g th
e on
es w
e di
scov
ered
in th
e re
adin
g. H
int:
The
re a
re s
ix o
f the
m.
7.)
Now
writ
e ou
t all
of th
e re
cipr
ocal
iden
titie
s. H
int:
The
re a
re s
ix o
f the
m.
8.)
Eval
uate
the
follo
win
g.
(A)
sin
23
0°
(B)
cos2
30
° (C
) ta
n2
30
° (D
) co
s24
5°
(E)
tan
24
5°
(F)
sin
26
0°
(G)
cos2
60
° (H
) ta
n2
60
° (I)
si
n( 3
0°2)
(J)
cos2𝛼
9.
) W
rite
out a
tabl
e of
val
ues f
or s
in2𝛼
,co
s2𝛼
, and
tan
2𝛼
, sta
rtin
g at
𝛼=
0, g
oing
up
by 5
° eac
h ro
w, a
nd e
ndin
g at
𝛼=
90
°. Yo
u w
ill n
eed
a ca
lcul
ator
for t
his
Exer
cise
. 10
.) U
se y
our r
esul
ts fr
om th
e pr
evio
us E
xerc
ise to
ans
wer
the
follo
win
g qu
estio
ns.
(A)
Wha
t is
the
max
imum
val
ue o
f sin
2𝛼
,co
s2𝛼
, and
tan
2𝛼
? (B
) W
hat i
s th
e m
inim
um v
alue
of s
in2𝛼
,co
s2𝛼
, and
tan
2𝛼
? (C
) Ar
e th
ere
any
simila
ritie
s or
diff
eren
ces
betw
een
sin
2𝛼
and
sin𝛼
? Co
mpa
re
your
resu
lts fr
om th
e pr
evio
us s
ectio
n.
11.)
One
of
the
mos
t im
port
ant
rela
tions
hips
in
Trig
onom
etry
is
the
Pyth
agor
ean
Iden
tity
we
disc
usse
d in
the
read
ing.
Writ
e th
is id
entit
y do
wn
now
. 12
.) Ev
alua
te th
e fo
llow
ing.
(A
) si
n2
60
°+
cos2
60
° (B
) co
s23
0°
+si
n2
30
° (C
) si
n2(𝛼
2)
+co
s2(𝛼
2)
(D)
sin
2( 3𝛼
+𝜋)
+co
s2( 3𝛼
+𝜋)
13.)
It is
ofte
n he
lpfu
l to
rew
rite
sin
2𝛼
or c
os2𝛼
. Use
the
Pyth
agor
ean
Iden
tity
to re
writ
e si
n2𝛼
and
co
s2𝛼
. 14
.) Si
mpl
ify th
e fo
llow
ing.
§3
Rel
atio
nshi
ps b
etw
een
the
Trig
func
tions
– 1
23
(A)
tan𝛼∙c
sc𝛼
(B
) ( s
in𝛼
+co
s𝛼)2
15
.) Ar
e th
ere
any
othe
r Pyt
hago
rean
Iden
titie
s? T
o fin
d th
is o
ut, u
se a
cal
cula
tor a
nd
try
the
follo
win
g fo
r diff
eren
t val
ues
of 𝛼
. (A
) se
c2𝛼
+cs
c2𝛼
(B
) ta
n2𝛼
+co
t2𝛼
(C
) Th
ere
are
two
othe
r Py
thag
orea
n Id
entit
ies.
Firs
t, us
ing
the
prev
ious
tw
o,
gues
s w
hat t
hey
mig
ht b
e. T
hen,
if y
ou c
an’t
figur
e it
out,
look
them
up
and
writ
e th
em d
own
now
. We’
ll di
scov
er h
ow to
arr
ive
at th
ese
resu
lts w
hen
we
have
som
e be
tter
tool
s. 16
.) An
swer
Tru
e or
Fal
se.
(A)
sin𝛼
=co
s(𝛼−
90
°)
(B)
sin
2𝛼
+co
s2𝛽
=1
iff 𝛼
+𝛽
=9
0°
(C)
sin
2𝛼
=si
n𝛼∙𝛼
(D
) si
n2𝛼
is s
omet
imes
neg
ativ
e.vi
vi A
ssum
e 𝛼∈ℝ
.
Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming.
April 27 - May 1, 2020 Course: Spanish III Teacher(s): Ms. Barrera [email protected] Supplemental links: www.conjuguemos.com www.spanishdict.com Weekly Plan: Monday, April 27 ⬜ Capítulo 5 - Read an announcement for a job position then choose the appropriate answer.. ⬜ Capítulo 5 - Translate your answers. Tuesday, April 28 ⬜ Capítulo 5 - Read the curriculum of Ms. Diaz and then choose the appropriate answer.. ⬜ Capítulo 5 - Translate your answers. Wednesday, April 29 ⬜ Capítulo 5 - Read the curriculum of Mr. Perez and then choose the appropriate answer.. ⬜ Capítulo 5 - Translate your answers. Thursday, April 30 ⬜ Capítulo 5 - Read about the history of Spain from the Celtiberos to the conquest of the muslims by Fernando e Isabela. ⬜ Capítulo 5 - Second, match and complete with the appropriate vocabulary according to your comprehension of the reading. Friday, May 1 ⬜ Capítulo 5 - Vocabulary Quiz A and B ⬜ Capítulo 5 - Vocabulary Quiz - C Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________Student Signature
I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature
Monday, April 27 Capítulo 5 - Read an announcement for a job position and then choose the appropriate answer. Then translate your answers for each number. A google document has been added in google classroom to type out your answers and submit within the lesson of the day. Please be organized and number your answers. I. Handout: Trabajo y comunidad: Lectura 1 - Read the Lectura and then choose the appropriate answer. II. Handout: Lectura 1 - Translate your answers. Tuesday, April 28 Capítulo 5 - Read Ms. Diaz’s Curriculum and then choose the appropriate answer. Then translate your answers for each number. A google document has been added in google classroom to type out your answers and submit within the lesson of the day. Please be organized and number your answers. I. Handout: Curriculum de Karen Díaz Manzanares: Lectura 2 - Read the Lectura and then choose the appropriate answer. II. Handout: Lectura 2 - Translate your answers. Wednesday, April 29 Capítulo 5 - Read the curriculum of Mr. Perez and then choose the appropriate answer. Then translate your answers for each number. A google document has been added in google classroom to type out your answers and submit within the lesson of the day. Please be organized and number your answers. I. Handout: Curriculum de Brandon Perez: Lectura 3 - Read the Lectura and then choose the appropriate answer. II. Handout: Lectura 3 - Translate your answers. III. Writing: Integration of ideas. Compare both curriculums. Then, according to your observations, then decide which of the two candidates is best prepared for the job from Lectura 1. Use information from the announcement and both curriculums to support your ideas. Thursday, April 30 Capítulo 5 - Read about the history of Spain from the Celtiberos to the conquest of the muslims by Fernando e Isabela. Second, match and complete with the appropriate vocabulary according to your comprehension of the reading. A google document has been added in google classroom to type out your answers and submit within the lesson of the day. Please when typing your answers label Exercise A, B, etc. I. Spanish Reading with Comprehension Exercises. La historia De España. Exercise A. Choose the appropriate letter. Exercise B. Choose the appropriate phrase in order to complete the sentence. Exercise C. Select the sentence that is false(falso) and rewrite it making it true. (cierto). For example, 1.
F then rewrite the sentence to make it true. Exercise D. Fill in the blank with the appropriate word from the reading. Friday, May 1 Capítulo 5 - Comprehension of work and volunteer vocabulary. I. Quiz 5-1: Comprensión del vocabulario 1. Type in your answers in the google doc provided for you in google classroom. A. Type in your 5 answers in the google doc. B. Type in your 8 answers in the google doc for this section. II. Quiz 5-1: Comprensión del vocabulario 1. Type in your answers in the google doc provided for you in google classroom for C. Because there are no numbers, please add them with your answers. There are 6 for the top and 6 for the bottom.
Physics Remote Learning Packet
May 4-8, 2020 Course: 11 Physics Teacher: Miss Weisse [email protected] Resource: Miss Weisse’s Own Physics Textbook — new pages found at the end of this packet Weekly Plan: Monday, May 4 ⬜ Read & Understand Notes on Unit 8 Part 3 – Hooke’s Law Background (pages 51-54) ⬜ Complete Pre-Lab Observations ⬜ Prepare Your “Lab Book” Entry Tuesday, May 5 ⬜ Take Data ⬜ Create 4 Graphs ⬜ Complete the Calculations and Analysis of All Four Graphs ⬜ Email Miss Weisse with Questions Wednesday, May 6 ⬜ Area Under the Curve Calculations for All Four Graphs ⬜ Explanation of the Physical Significance of the Area Under the Curve ⬜ Write Lab Conclusion ⬜ Email Miss Weisse with Questions and to Ask for Unit 8 Part 4 of Miss Weisse’s Own Physics Textbook Thursday, May 7 ⬜ Read & Understand Notes on Unit 8 Part 4 - Hooke’s Law & Elastic Energy Notes ⬜ Complete Unit 8 Worksheet 2 #1-4 ⬜ Email Miss Weisse with Questions and to Ask for Solutions Friday, May 8 ⬜ Attend Office Hours at 9:30am! ⬜ Review Unit 8 Part 4 - Hooke’s Law & Elastic Energy Notes (pages 55-57) ⬜ Complete Unit 8 Worksheet 2 #5-7 ⬜ Email Miss Weisse with Questions and to Ask for Solutions Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________ Student Signature
I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature
Monday, May 4
➔ Read & Understand Notes on Unit 8 Part 3 – Hooke’s Law Background (pages 51-54)
➔ Complete Pre-Lab Observations
◆ Watch the short, two-minute video (https://safeYouTube.net/w/wMh8), observe the image of four spring scales below, and image the behavior of springs or rubber bands to answer the following three familiar questions on a sheet of paper, or on a google doc, to be turned in.
● What do you see?
● What can you measure?
● What can you change?
**As you observe the four spring scales, describe both the motion and behavior of the springs as they move and the difference between the four springs.
➔ Prepare Your “Lab Book” Entry on a piece of paper. This assignment cannot be completed online.
◆ Write the Purpose of the lab (can be found in Unit 8 – Part 3... if you don’t already know)
◆ List your Independent Variable (length of spring), Dependent Variable (force of spring), and Constants (determine these yourself).
*** Are our variable in your list of things we can measure and change?!
◆ Create four data charts like the example chart below.
NOTE! Extension is the length the spring stretched when a force is applied, not the total length of the spring. To calculate this, subtract the length at 0 Newtons force from each measure length.
Extension = Current Length – Length @ 0 Newton Force
Green Scale Force (N) Length (cm) Extension (cm) Extension (m)
Tuesday, May 5
➔ Take data using the images found at the end of this packet, after the textbook.
◆ Don’t forget our precision measurement rules! You should write a measurement that has one more decimal place than the ruler marks. So each of your measurements should have 2 decimal places. Also, to take good data, zoom in on your screen to ~200%, then use a piece of paper to create a line between the bottom of the spring and the ruler markings.
➔ Create four graphs from your four sets of data. Each graph should take up at least ⅓ - ½ of the page.
➔ Complete the calculations and analysis for each of the four graphs
As a reminder –
◆ The calculations include the slope, the y-intercept, and the 5% rule for the y-intercept if your value is not zero.
◆ The analysis includes what the slope tells us (focus on units!) and what the y-intercept tells us.
◆ Some helpful reminders are found on page 54 of Miss Weisse’s Own Physics Textbook
Wednesday, May 6
➔ Now that you have an equation that describes the relationship between force exerted by the spring and the extension of the spring, it is time to relate this information to ENERGY. If you recall from our study of motion 1st semester, we found new, important physical properties by calculating the Area Under the Curve (AUC) of the velocity versus time graph and acceleration vs. time graph. We can do the same thing here with our spring force versus length graph. Calculate the AUC for all four graphs.
➔ Explain the physics significance of the Area Under the Curve. What physical quantity does this measurement give us? Use the units to help puzzle it out.
➔ Write a Lab Conclusion.
◆ Restate the problem and summarize how the data was taken. ◆ State the relationship between the two variables, explain what the slope means, and state what the
y-intercept tells us. ◆ Compare the four different springs, giving numerical data. ◆ Explain what the area under the curve tells us. ◆ If your data has problems, provide possible explanations. ◆ Provide an answer to the original problem.
***Today’s work can be done on the same piece of paper as Tuesday’s lab work.
➔ Email Miss Weisse with Questions and to Ask for Unit 8 Part 4 of Miss Weisse’s Own Physics Textbook
Thursday, May 7
➔ If you have not done so yet, turn in your lab and request Unit 8 Part 4 from Miss Weisse via Email!
➔ Read & Understand Notes on Unit 8 Part 4 - Hooke’s Law & Elastic Energy
➔ On a sheet of paper with a full heading, complete Unit 8 Worksheet 2 #1-4
Energy Storage and Transfer Model Worksheet 2 (part #1-4): Hooke’s Law and Elastic Energy
Directions: Suppose one lab group found that F = 1000 N/m (∆x). Construct a graphical representation of force vs. displacement. (Hint: make the maximum displacement 0.25 m. )
1. Graphically determine the amount of energy stored while stretching the spring described above from x = 0 to x = 10. cm. 2. Graphically determine the amount of energy stored while stretching the spring described above from x = 15 to x = 25 cm.
Directions: The graph below was made from data collected during an investigation of the relationship
between the amounts two different springs stretched when different forces were applied. 3. Determine the spring constant for each spring.
4. For each spring, compare: a. the amount of force required to stretch the spring 3.0 m. b. the Eel stored in each spring when stretched 3.0m.
Friday, May 8
➔ Attend Office Hours at 9:30am! I want to talk to you and help you!
➔ Review Unit 8 Part 4 - Hooke’s Law & Elastic Energy Notes
➔ On a sheet of paper with a full heading, complete Unit 8 Worksheet 2 #5-7
Energy Storage and Transfer Model Worksheet 2 (#5-7): Hooke’s Law and Elastic Energy
5. Determine the amount that spring 2 needs to be stretched in order to store 24 joules of energy. 6. The spring below has a spring constant of 10. N/m. If the block is pulled 0.30 m horizontally to the
right, and held motionless, what force does the spring exert on the block? Sketch a force diagram for the mass as you hold it still. (Assume a frictionless surface.)
7. The spring below has a spring constant of 20. N/m. The µs between the box and the surface is 0.40.
a. The box is pushed to the right, then released. Draw a force diagram for the box above when the spring
is stretched, yet the box is stationary. b. What is the maximum distance that the spring can be stretched from equilibrium before the box begins
to slide back? c. Do pie chart analysis for this situation, when the spring is stretched beyond its maximum (from part b
above) so it slides back, and then the box oscillates back and forth until it comes to a stop. Assume your system includes the spring, box, and table top.
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SET UP ********************to make your measurements you will want to zoom in on your computer screen (~200%)********************
FROM THE FRONT: The spring scale is being anchored in place with a pen perpendicular to the board, so the whole scale cannot be pulled forward when the hook is pulled forward to extend the spring.
FROM THE TOP: Again, the spring scale is anchored by the pen and the top of the spring is aligned with the zero mark of the ruler. The ruler remains stationary throughout the measurements.
GREEN SPRING SCALE (VIDEO)
WHITE SPRING SCALE (VIDEO)
RED SPRING SCALE (VIDEO)
BROWN SPRING SCALE (VIDEO)