Unit R Student Success Sheet (SSS) - Santa Ana …. Sin(405o+120o) vs. Sin405o + sin 120o 10. Sin()...

Unit R Student Success Sheet (SSS) Trigonometric Identities Part 2 (section 5.4)

Standards: Trig 10.0

Segerstrom High School -- Math Analysis Honors

© Crystal Kirch 2011

Name: ____________________________________

Period: __________

Thinkbinder Study Group: www.bit.ly/ChatUnitR

Reminders:

Practice Problems (PQ & PT) are completed in spiral bound notebook only.

All pages in spiral notebook should be labeled accordingly:

Unit ______ Concept ______ - (title of assignment) Examples:

Unit R Concept 1 – Practice Quiz Unit R Concept 1-4 – Practice Test

Website with all video links and resources:

kirchmathanalysis.blogspot.com Edmodo Group Codes for class communication:

http://bit.ly/edmodo2013

“I find that when you have a real interest in life and a curious life,

that sleep is not the most important thing.”

Martha Stewart

Concept # What we will be learning… Mandatory

Practice Optional Extra practice

from textbook

1 Exact value of sums or differences (might have to find what it adds or subtracts to)

Practice quiz 1

2 Using sum and difference formulas when given values (right triangles)

Practice quiz 2

3 Trig of an inverse trig function (use sum and difference to prove)

Practice quiz 3

4 Simplifying sum and difference identities Practice quiz 4

5 Solving trig equations with sum and difference Practice quiz 5

We will be continuing our study of trigonometric identities, focusing on sum and difference identities. These identities are most helpful to us when we want to find exact values of angles that AREN’T on the unit circle, but can be created by either adding or subtracting two angles that ARE on the unit circle! So, this expands our ability to use the unit circle for just 0,30,45,60, and 90 degree reference angles to many more!

---Unit R Student Success Sheet--- Trigonometric Identities Part 2 (section 5.4)--- Math Analysis Honors---


Exact value of sums or differences (might have to find what it adds or subtracts to)

Sometimes we want to find the exact values of angles that AREN’T on the UNIT CIRCLE. This concept will show us that we can find the exact value of any angle of which angles on the unit circle can ADD (sum) or SUBTRACT (difference) to that angle!

We must first realize what this DOESN’T mean…



For the following problems, find the exact values (this means your answer will have RADICALS in it and NOT DECIMALS!) of the sine, cosine, and tangent of the angle. We must first decide what angles to use to either add or subtract to the one we want. There is often MORE THAN ONE OPTION for this!

Sin Cos tan

1. 105o = 60o + 45o U ordered pair:

cos =

sin=

tan=

V ordered pair:

cos =

sin=

tan=

u= _______ v= ________

u= _______ v= ________ u= _______ v= ________

Sin Cos Tan

2. 75o = U ordered pair:

cos =

sin=

tan=

V ordered pair:

cos =

sin=

tan=

u= _______ v= ________

u= _______ v= ________ u= _______ v= ________



3.

=

U ordered pair:

cos =

sin=

tan=

V ordered pair:

cos =

sin=

tan=

u= _______ v= ________

u= _______ v= ________ u= _______ v= ________

4. -225o= U ordered pair:

cos =

sin=

tan=

V ordered pair:

cos =

sin=

tan=

u= _______ v= ________

u= _______ v= ________

u= _______ v= ________

Sin Cos Tan

5.

=

U ordered pair:

cos =

sin=

tan=

V ordered pair:

cos =

sin=

tan=

u= _______ v= ________

u= _______ v= ________

u= _______ v= ________



Write the expression as either the sine, cosine, or tangent of an angle

6. cos60ocos20o – sin60osin20o

7.

8.

Additional Problems on “extra videos” Extra video #1 covers the following problems. Use these as extra practice and the videos as a guide if you need help.

9. Sin(405o+120o) vs. Sin405o + sin 120o 10. Sin(

) vs. Sin(

)

Extra video #2 covers the following problems. Use these as extra practice and the videos as a guide if you need help.

Find the sine, cosine, and tangent of the following angles 11. 165o = 135o + 30o 12. 15o 13.

PQ problems:

Find the sine, cosine, and tangent of the following angles

14.

15.

16.

17. 18. 19.

Write the expression as either the sine, cosine, or tangent of an angle 20. sin110ocos80o +cos110osin80o (these are in radians!)

21. sin3.5cos1.2 – cos3.5sin1.2 22.

PT problems:

Find the sine, cosine, and tangent of the following angles

23.

24.

25.

26.

Write the expression as either the sine, cosine, or tangent of an angle

27.

28. (these are in radians!)

Cos(.96)cos(.42) + sin(.96)sin(.42) 29.



Using sum and difference formulas when given values (right triangles)

Begin by : Drawing the appropriate triangles in the correct quadrants.

Find the missing piece of the triangle. BE CAREFUL OF THE SIGNS (left and down is negative, right and up is positive!)

Plug into the sum and difference formulas.

1. Find the exact value of the trigonometric function given that

and

. (Both u and v are in Quadrant II)

Sin u =

Cos u =

Tan u =

Sin v =

Cos v =

Tan v =

Sin (u+v) cos (u+v) tan (u+v)

Sin (u – v) Cos (u – v) Tan (u – v)

Sin (v+u) cos (v+u) tan (v+u)

Sin (v – u)

Cos (v – u) Tan (v – u)



2. Find the exact value of the trig function given that

and

. (Both u and v are in Quadrant III)

Sin u =

Cos u =

Tan u =

Sin v =

Cos v =

Tan v =

Sin (u+v) cos (u+v) tan (u+v)

Sin (u – v) cos (u – v) tan (u – v)

Sin (v+u) cos (v+u) tan (v+u)

Sin (v – u) cos (v – u) tan (v – u)




3. Cos(v – u) for #1 above 4. Tan(u + v) for #2 above

PQ problems: *Please note the varying ways the problem/quadrants could be given!

5.

6.

7.

8.

9.

10.

PT problems: *Please note the varying ways the problem/quadrants could be given!

11.

12.

13.

14.

15.

16.



Trig of an inverse trig function (use sum and difference to prove)

This concept has THREE types of problems, all solved in a very similar way…

Type 1 Directions: Write the trigonometric expression as an algebraic expression.

1. *arcsin is another notation for sin-1x

Write like sin(u+v) and identify u and v

U= __________________ V= __________________

Take the trig function of each side to get rid of the inverse trig function

U=arcsinx turns into _________________ V=arccosx turns into_________________

Draw a triangle for each piece and label it appropriate with the given information. Use Pythagorean Theorem as

needed. It’s okay for there to be variables still in the problem!!!

Use the sum and difference formulas to simplify

Type 2 Directions: Find the value of the expression without using a calculator.

2. Write like sin(u+v) and identify u and v U= __________________ V= __________________

Use your knowledge of the Unit Circle to identify what u and v actually are.

Plug into sum and difference formulas and simplify.

3.

4.



Type 3 Directions: Use Right triangles to evaluate the expression.

5. Write like sin(u+v) and identify u and v U= __________________ V= __________________

Take the trig function of each side to get rid of the inverse trig function

U=cos-1 3/5 turns into _________________ V= sin-1 5/13 turns into_________________

Draw a triangle for each piece and label it appropriate with the given information. Use Pythagorean Theorem as

needed.

Use the sum and difference formulas to simplify


6. cos(arccos x – arcsin x)


7. 8. 9.


10.

]

PQ problems: Evaluate the expression.

11. 12. 13.

14. 15.

16. 17.

PT problems: Evaluate the expression.

18. 19. 20.

21. 22.



Simplifying sum and difference identities

Expand the expression using sum and difference identities.

Combine like terms and simplify.

1.

2.

3.

4.




5. 6.

PQ problems: Simplify or verify the trigonometric identity.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

PT problems:

17. 18.

19.

20.

21. 22.

23.

24. and * Mrs. Kirch couldn’t figure this one out when she tried… if you can, you get extra credit! Turn in your work page to the box as soon as possible. Extra credit is limited to the first person from each class period to submit!



Solving trig equations with sum and difference

Expand the expression using sum and difference identities. Combine like terms and simplify as much as you can.

Solve the following equation from [0,2π)

1. cos (x + π/4) + cos (x – π/4) = 0

2.


3.

PQ problems: Solve the equation from [0,2π)

4.

PT problems: Solve the equation from [0,2π)

5.

Collaborative Answer Key will be posted on kirchmathanalysis.blogspot.com

Unit R Student Success Sheet (SSS) - Santa Ana …. Sin(405o+120o) vs. Sin405o + sin 120o 10. Sin()...

Documents

Transcript of Unit R Student Success Sheet (SSS) - Santa Ana …. Sin(405o+120o) vs. Sin405o + sin 120o 10. Sin()...