Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6...

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Lesson 13.4, For use with pages 875-880 1. cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 6 3. tan(– 60º) ANSWER 3 ANSWER 2 2

Transcript of Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6...

Page 1: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

Lesson 13.4, For use with pages 875-880

1. cos 45º

ANSWER1

2

Evaluate the expression.

2. sin 5π

6

3. tan(– 60º)

ANSWER – 3

ANSWER2

2

Page 2: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

ANSWER –1

Lesson 13.4, For use with pages 875-880

ANSWER3

3–

5. tan – π

6

Evaluate the expression.

4. cos π

Page 3: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

Trigonometry, Inverse Functions

Page 4: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

EXAMPLE 1 Evaluate inverse trigonometric functions

Evaluate the expression in both radians and degrees.

a. cos–1 3

2

SOLUTION

a. When 0 θ π or 0° 180°, the angle whose cosine is

≤ ≤ ≤ θ ≤ 3

2

cos–1 3

2

√θ =

π

6= cos–1 3

2

√θ = = 30°

Page 5: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

EXAMPLE 1 Evaluate inverse trigonometric functions

Evaluate the expression in both radians and degrees.

b. sin–1 2

SOLUTION

sin–1b. There is no angle whose sine is 2. So, is undefined.

2

Page 6: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

EXAMPLE 1 Evaluate inverse trigonometric functions

Evaluate the expression in both radians and degrees.

3 ( – c. tan–1 √

SOLUTION

c. When – < θ < , or – 90° < θ < 90°, the

angle whose tangent is - is:

π

2

π2

√ 3

( – )tan–1 3√θ =π

3–= ( – )tan–1 3 √θ = –60° =

Page 7: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

EXAMPLE 2 Solve a trigonometric equation

Solve the equation sin θ = – where 180° < θ < 270°.5

8

SOLUTION

STEP 1

sine is – is sin–1 – 38.7°. This5

8

5

8–

Use a calculator to determine that in the

interval –90° θ 90°, the angle whose≤ ≤

angle is in Quadrant IV, as shown.

Page 8: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

EXAMPLE 2 Solve a trigonometric equation

STEP 2

Find the angle in Quadrant III (where

180° < θ < 270°) that has the same sine

value as the angle in Step 1. The angle is:

θ 180° + 38.7° = 218.7°

CHECK : Use a calculator to check the answer.

5

8sin 218.7° – 0.625 = –

Page 9: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

GUIDED PRACTICE for Examples 1 and 2

Evaluate the expression in both radians and degrees.

1. sin–1 2

2

ANSWERπ4

, 45°

2. cos–1 12

ANSWERπ3

, 60°

3. tan–1 (–1)

ANSWERπ4

, –45°–

Page 10: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

GUIDED PRACTICE for Examples 1 and 2

Evaluate the expression in both radians and degrees.

4. sin–1 (– )12

π6

, –30°–ANSWER

Page 11: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

GUIDED PRACTICE for Examples 1 and 2

Solve the equation for

270° < θ < 360°5. cos θ = 0.4;

ANSWER about 293.6°

180° < θ < 270°6. tan θ = 2.1;

ANSWER about 244.5°

270° < θ < 360°7. sin θ = –0.23;

ANSWER about 346.7°

Page 12: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

GUIDED PRACTICE for Examples 1 and 2

Solve the equation for

180° < θ < 270°8. tan θ = 4.7;

ANSWER about 258.0°

90° < θ < 180°9. sin θ = 0.62;

ANSWER about 141.7°

180° < θ < 270°10. cos θ = –0.39;

ANSWER about 247.0°

Page 13: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

EXAMPLE 3 Standardized Test Practice

SOLUTION

In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse, so use the inverse cosine function to solve for θ.

cos θ =adj

hyp=

6

11cos – 1θ =

6

1156.9°

The correct answer is C.ANSWER

Page 14: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

EXAMPLE 4 Write and solve a trigonometric equation

Monster Trucks

A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle θ of the ramp?

Page 15: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

EXAMPLE 4 Write and solve a trigonometric equation

SOLUTION

STEP 1 Draw: a triangle that represents the ramp.

STEP 2 Write: a trigonometric equation that involves the ratio of the ramp’s height and horizontal length.

tan θ =opp

adj=

8

20

Page 16: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

EXAMPLE 4 Write and solve a trigonometric equation

STEP 3 Use: a calculator to find the measure of θ.

tan–1θ = 8

2021.8°

The angle of the ramp is about 22°.

ANSWER

Page 17: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

GUIDED PRACTICE for Examples 3 and 4

Find the measure of the angle θ.

11.

SOLUTION

In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse. So, use the inverse cosine function to solve for θ.

cos θ =adj

hyp=

49

= 63.6°θ cos–1 49

Page 18: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

GUIDED PRACTICE for Examples 3 and 4

Find the measure of the angle θ.

SOLUTION

In the right triangle, you are given the lengths of the side opposite to θ and the side adjacent. So, use the inverse tan function to solve for θ.

12.

tan θ =opp

adj=

108

θ 51.3°= tan–1 108

Page 19: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

GUIDED PRACTICE for Examples 3 and 4

Find the measure of the angle θ.

SOLUTION

In the right triangle, you are given the lengths of the side opposite to θ and the hypotenuse. So, use the inverse sin function to solve for θ.

13.

sin θ =opp

hyp=

512

24.6°θ = sin–1 512

Page 20: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

GUIDED PRACTICE for Examples 3 and 4

14. WHAT IF? In Example 4, suppose a monster truck drives 26 feet on a ramp before jumping onto a row of cars. If the ramp is 10 feet high, what is the angle θ of the ramp?

SOLUTION

STEP 1 Draw: a triangle that represents the ramp.

STEP 2 Write: a trigonometric equation that involves the ratio of the ramp’s height and horizontal length.

tan θ =opp

adj=

10

26

Page 21: Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6 3.tan(– 60º) ANSWER – 3 ANSWER 2 2.

GUIDED PRACTICE for Examples 3 and 4

STEP 3 Use: a calculator to find the measure of θ.

22.6°tan–1θ =10

26

The angle of the ramp is about 22.6°.

ANSWER