3.3 Trigonometry Name ________________________

The first part of this set of lecture notes will be considered an activity.

Trigonometry is the study of right triangles. More specifically, it is the study of how angles and sides relate to each other. Before we make use of these relationships we are going to first determine them. You will need a calculator, protractor, ruler, pencil and (of course) an eraser.

Fill out Table 1 by measuring each side to the nearest 10th of a centimeter. Measure each angle to the nearest degree.#1

NOTE: In these exercises we will use the protocol that angle C is the right angle and “c” is the hypotenuse, side “a” is opposite angle A, and side “b” is opposite angle B.

Triangle 1 Triangle 2

1

A

c b

C B a

A

c b

C B a

The results are not going to be PERFECT because of errors that accumulate due to drawing with computer tools and reproduction differences.

* I am looking for a decimal answer rounded off to two decimal places.

1) If all went well, you should find the two triangles to be “similar.” That means, they are the same shape, just a different size. Angles A, B and C should be the same (or very, very, very close – within a degree or two) in both triangles. If this is not true, you should go back and check.

2) Also, given that the triangles ARE similar, the ratios you found (not the values for a, b and c) should be the same (or very, very, very close) as well.

If your values are “way off,” go back and measure again.

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TABLE #1 Triangle 1 Triangle 2

Side a (cm)

Side b (cm)

Side c (cm)

Angle A

Angle B

Angle C

*Ratio of a to c

*Ratio of b to c

*Ratio of b to a

It is important for you to know that it is the measures of the given angles and sides that drive the measures of the angles and sides that

you don’t know. Let me explain by example.#2At the bottom of the page, draw a right triangle labeled ABC with the following criteria:

a)

b)

c) Let leg “a” = 6 cm (the side opposite ).

Find the measure of angle C by actually measuring it. Fill in Table 2.

Find the measure of the missing leg “b” – it is opposite angle B. Fill in Table 2.

Find the measure of the hypotenuse “c” – it is opposite angle C. Fill in Table 2.

Check your answers by using the Pythagorean Theorem.

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TABLE #2 GIVEN 2 ANGLES AND 1 SIDE

Angle A

Angle B 30o

Angle C 90o

Side (leg) a 6cm

Side (leg) b

Side (hyp) c

#3 At the bottom of the page, draw a right triangle labeled ABC with the following criteria:

a)

b) Side a (opposite angle A) = 8 cm

c) Side b (opposite angle B) = 9 cm

Find the measure of angles B and C by actually measuring them. Do they add to 90o? They should. Fill in Table 2.

Find the measure of side “c” (the hypotenuse) by actually measuring it. Check your numbers with the Pythagorean Theorem. Fill in Table 2.

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TABLE #3SIDE ANGLE SIDE

Angle A

Angle B

Angle C 90o

Side (leg) a 8cm

Side (leg) b 9cm

Side (hyp) c

The message is: If you know

two angles and one side or

two sides and one anglethen you can find the missing sides and missing angles.

This is where the trigonometry comes in. Recall in a previous table, we looked at some ratios. Remember, these ratios were driven by the angles in the triangle (and, of course, the

sides). Trigonometry uses a shortcut method of referring to these ratios. The shortcut

terminology is sine, cosine, and tangent. Each of these means a very specific ratio.

Look at the triangle below. Find the angle Θ (which is the same as angle B).

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The side ADJACENT to Θ is “a.:The side OPPOSITE Θ is “b.”The HYPOTENUSE is “c.”

Consider angle A(stand on it)

The ratio of a/c is the

cosine of Θ

The ratio of b/c is the sine of Θ

The ratio of b/a is the

cosine of Θ

Now, do the same for angle A, use this column.

Draw another right triangle. Label the sides r, s, t. Name the triangele RST. So we are all looking at the same kind of triangle, make S the right angle. Fill out the table.

Angle R Angle T

Find the side adjacent _______________ _______________

Find the side opposite _______________ _______________

Find the hypotenuse _______________ _______________

Now, we’re ready to apply the trig functions to what they represent.It all boils down to this:

SOH – CAH – TOA

Pronounced: “so-cuh-toe-uh”

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I know it sounds kooky, but THIS you MUST memorize.Practice:Draw a right triangle – any right triangle.

Name the angles (you pick) and assign values to all 3 angles, remember, this is a right triangle.

Name the Someone in class needs to choose.a. adjacent side ___________

b. opposite side ___________

c. hypotenuse ___________

Fill in the measures of the angles. Then, use your calculator to fill in the rest of the table. Round your answers off to the nearest tenth.

Angle ____ = _______° cos _____ = ______ sin _____ = ______ tan _____ = ______

Angle ____ = _______° cos _____ = ______ sin _____ = ______ tan _____ = ______

Angle ____ = _______° cos _____ = ______ sin _____ = ______ tan _____ = ______

Need more explanation? Check out this website a student found. The explanations are VERY straightforward. (Thanks, M.B.)http://www.youtube.com/watch?v=T8B8Tcz8kog&feature=related

And this one from Khan Academyhttp://www.khanacademy.org/math/trigonometry/basic-trigonometry

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http://www.khanacademy.org/math/trigonometry/basic-trigonometry

Time to learn how to use your calculator for trig functions.

Example 1: Find the missing sides. Round to the hundredths.We know two angles and one side (the hypotenuse.)

Find x (the side adjacent to ).Which function should we use?

Find y (the side opposite ).Which function should we use?

Do these numbers work using the Pythagorean Theorem?

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Example 2: Find the missing sides to the nearest 16th inch.We know two angles (90 and ?) and one side (leg).

Find w.Which function should we use?

Find z.Which function should we use?

Do these numbers work using the Pythagorean Theorem?

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Example 3: Find the horizontal and vertical distance to drill holes in locations A and B. We are looking for coordinates (x,y) for each location.

Start with A.

B?

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Recall that we use these functions when you know two angles (90 and ?) and one side.

Now, we will look at three new functions. All are related to the three trig functions we know.

These new functions are called INVERSE TRIG functions.

They look like this:

We use these functions when we are looking for an ANGLE and we KNOW two sides.

Check out these videos:http://youtu.be/pZDsiOSUznk

Example 4: Find angles A and B in a roof sloped at 7/12. Round to the nearest tenth place.

A?

B?

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Example 5: Finding the angles in the steel tubing.Welders often use angled braces to add strength and rigidity to a weld. Find angles A and B and the length of the tubing, from long point to long point, from the given dimensions.

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Homework: You are assigned all of the problems from 1 - 23. Take a look at problems 24 – 27. If you are in the manufacturing field, you REALLY need to be able to do these. I will give you an additional 2 points each (to be added to your next test), if you do them correctly.

Practice 1: Figure out the angles and lengths in a regular pentagonQ1. Find the length

of one sideQ2. Find the total

heightQ3. Find the total

widthQ4. Find the length

of a diagonal

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Practice 1 Solved: Figure out the angles and lengths in a regular pentagonQ1. Find the length of one sideQ2. Find the total heightQ3. Find the total widthQ4. Find the length of a diagonal (????)

Three triangles (totally 540⁰), divided by 5angles = 108 ⁰ for each outside angle.The interior angles as shown below are each 36⁰.

Q1: length of one side (2 times ”a”)

Length of one side is 14.10cm

Q2: total height “b” plus the radius (12cm).

Total height is 21.71cm.

Q3: total width (2 times “c”)

Total width is 11.41cm.

Q4: length of a diagonal

14

c

12cm 72⁰

c

36⁰12cm

54⁰ b 54⁰

a

Practice 2: Figure out the horizontal and vertical distance to each point.Q1. Point AQ2. Point BQ3. Point C

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Practice 2 Solved: Figure out the horizontal and vertical distance to each point. Round to the hundredth.Q1. Point AQ2. Point BQ3. Point C

Point A. Focus on a right triangle that can be drawn which includes point A.

If we can find the measure of angle 1,we can find the lengths of the twosides (a and b).

First find the angular distance between each of the points. Therefore,

Thus, the horizontal and vertical distance to point A is (35.96mm, 28.68mm). Check with Pythagorean theorem.

Point B. Focus on a right triangle that can be drawn which includes point B.Find the length of side c.

Find the length of side d.

Thus, the horizontal and vertical distance to point B is (44.85mm, -10.24mm). Check with Pythagorean theorem.Why is the vertical position negative?????

Point C. Focus on a right triangle that can be drawn with includes point C.Find the length of side f. Find the length of side g.

Thus, the horizontal and vertical distance to point C is (-19.96mm, -41.45mm).Check with Pythagorean theorem. Why are both the horizontal and vertical positions negative?????

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A 46mm

a1 b

A 46mm

1 c

angle 2 46mm side d

angle 3

f

g 46mm

C

Practice 3. Figure out the horizontal and vertical distance to each point.Q1. Point A Q4. Point DQ2. Point B Q5. Point EQ3. Point C Q6. Graph the points on graph paperPractice 3 Solved. Figure out the horizontal and vertical distance to each point.Q1. Point A Q4. Point DQ2. Point B Q5. Point EQ3. Point C Q6. Graph the points on graph paper

Point A. Determine the measure of each angle in a pentagon.Three triangles (totally 540⁰), divided by 5angles = 108⁰ each.

18⁰ Find the length of side a.Point E. Use the same triangle as for point A.

18” Find the length of side b.side b (0, 17.12”)

108⁰

side a (5.56”, 0) Use Pythagorean Theorem to check. 180-108=72⁰

Point B. Add 18” to the value from point A. Point C. Add “a” to the value from point B.go get the horizontal. Use “b” for the vertical.

(23.56”, 0)(29.12”, 17.12”)

Point D. First find sides “c” and “d.”

The horizontal for point D is “d” (14.56in). The vertical for point D is “b” plus “c” (17.12+10.58=27.7in).

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d D d

c c

E C

b b

a A 18” B a

side d

side c

54⁰

108⁰18⁰

(14.56in, 27.7in)

Section 3.3:

1. A = 85.2o

B = 4.8o

2. guy wire ≈ 30.6 ft and d ≈ 14.4 ft3. 41.4o

4. 37.5o

5. 11 5”16

6. 76o

7. C = 236 7”16 or 19’ - 8

7”16

D = 132 3”16 or 11’ -

3”16

8. W = 15.7 mH = 8.7 m

9. A = 132.1o, B = 47.9o, C = 42.1o, D = 18.9o, E = 411.6 m, F = 268.7 m, G = 199.6 m

10. H = D = 33 15”16 or 2’ - 9

15”16

11. 849 mm12. x = 25.7 cm, y = 8.3 cm13. x = 125.8 mm, y = -173.1 mm14. A = 32.9o

B = 57.1o

h = 13 1”8

15. x = 2 11”16 , y = 1

7”8

16. 2 9”16

17. A = (96.5 mm, 17.0 mm)B = (-63.0 mm, 75.1 mm)C = (-33.5 mm, -92.1 mm)D = (84.9 mm, -49.0 mm)

18. A = 25.5o

B = 64.5o

C = 232.6 cm

19. A = 33.7o

B = 56.3o

C = 81 3”4

D = 6 5”8

20. L ≈ 5.56T ≈ 3,336 ft-lbs (using the rounded value for L)

21. 45 1”2 , 43, 40

1”2 , 38, 35

1”2

22. 1248” (1249” if you don’t round until the end)23. 37o

24.

Bend Allowance (B) Set Back (D).962 mm .676 mm

33”8 2

3”16

25. a) 4"b) 7.35"

26. a) 5.482"b) 9.781"

27. a) 4.515 cmb) 6.495 cm

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