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

2. –1 Lesson 13.4, For use with pages 875-880 ANSWER 3 3 – 5. tan – π 6 Evaluate the expression. 4.cos π

3. Trigonometry, Inverse Functions

4. 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 = 3 2 √ θ = = 30°

5. EXAMPLE 1 Evaluate inverse trigonometric functions Evaluate the expression in both radians and degrees. b.sin –1 2 SOLUTION sin –1 b. There is no angle whose sine is 2. So, is undefined. 2

6. 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 π 2 √3 ( – ) tan –1 3√θ = π 3 – = ( – ) tan –1 3√θ = –60° =

7. 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°. This 5 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.

8. 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 8 sin 218.7°– 0.625 = – 

9. GUIDED PRACTICE for Examples 1 and 2 Evaluate the expression in both radians and degrees. 1.sin –1 2 2 √ ANSWER π 4, 45° 2.cos –1 1 2 ANSWER π 3, 60° 3.tan –1 (–1) ANSWER π 4, –45° –

10. GUIDED PRACTICE for Examples 1 and 2 Evaluate the expression in both radians and degrees. 4.sin –1 (– ) 1 2 π 6, –30° – ANSWER

11. 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°

12. 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°

13. 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 11 cos – 1 θ = 6 11 56.9° The correct answer is C. ANSWER

14. 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?

15. 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

16. EXAMPLE 4 Write and solve a trigonometric equation STEP 3 Use: a calculator to find the measure of θ. tan –1 θ = 8 20 21.8° The angle of the ramp is about 22°. ANSWER

17. 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 = 4 9 = 63.6° θcos –1 4 9

18. 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 = 10 8 θ 51.3° = tan –1 10 8

19. 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 = 5 12 24.6° θ = sin –1 5 12

20. 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

21. GUIDED PRACTICE for Examples 3 and 4 STEP 3 Use: a calculator to find the measure of θ. 22.6° tan –1 θ = 10 26 The angle of the ramp is about 22.6°. ANSWER