9.4 Evaluate Inverse Trigonometric Functions

1. 9.4 Evaluate Inverse Trigonometric Functions How are inverse Trigonometric functions used? How much information must be given about side lengths in a right triangle in order for you to be able to find the measures of its acute angles?

3. Inverse Trig Functions y x

4. Inverse Trig Functions y x 0

5. Inverse Trig Functions y x

6. √ √ √ √ 3 3 3 3 a. cos–1 2 2 2 2 When0θπor0°180°,the angle whose cosine is ≤ ≤ ≤ θ ≤ a. π 30° cos–1 = θ = 6 cos–1 = θ = Evaluate the expression in both radians and degrees. SOLUTION

7. y 90° 120° 60° 45° 135° 30° 150° 0° 360° 180° x 330° 210° 315° 225° 240° 300° 270°

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

9. ( – ) √ √ √ √ c. tan–1 3 3 c. When –< θ < , or – 90°< θ < 90°, the angle whose tangent is – is: – ( – ) ( – ) –60° = = θ tan–1 3 θ tan–1 3 = = π π π 2 3 2 Evaluate the expression in both radians and degrees. SOLUTION

10. √ 2 2. 1. 3. 4. sin–1 tan–1 (–1) cos–1 sin–1 (– ) 2 , 45° ANSWER π π π π , –30° ANSWER – 6 4 3 4 1 1 2 2 , 60° ANSWER , –45° ANSWER – Evaluate the expression in both radians and degrees.

11. Solve the equationsinθ = – where180° < θ < 270°. 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: ≤ ≤ interval –90° θ 90°, the angle whose Use a calculator to determine that in the 180° + 38.7° = 218.7° θ 5 5 5 5 –  = sin 218.7° – 0.625 8 8 8 8 – sine is – is sin–1– 38.7°. This angle is in Quadrant IV, as shown. Solve a Trigonometric Equation SOLUTION STEP 1 STEP 2 Use a calculator to check the answer. CHECK :

12. ANSWER ANSWER ANSWER about 293.6° about 244.5° about 346.7° Solve the equation for 5. cos θ = 0.4; 270° < θ < 360° 6. tan θ = 2.1; 180° < θ < 270° 7. sin θ = –0.23; 270° < θ < 360°

13. ANSWER ANSWER ANSWER about 258.0° about 141.7° about 247.0° Solve the equation for 8. tan θ = 4.7; 180° < θ < 270° 9. sin θ = 0.62; 90° < θ < 180° 10. cos θ = –0.39; 180° < θ < 270°

14. The correct answer is C. ANSWER 6 6 adj 56.9° θ cos – 1 = cosθ = = 11 11 hyp 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.

15. 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? http://www.youtube.com/watch?v=7SjX7A_FR6g http://www.youtube.com/watch?v=SrzXaDFZcAo

16. Draw: a triangle that represents the ramp. ANSWER The angle of the ramp is about 22°. 8 opp tanθ = = 20 adj 8 21.8° θ tan–1 = 20 SOLUTION STEP 1 Write: a trigonometric equation that involves the ratio of the ramp’s height and horizontal length. STEP 2 STEP 3 Use: a calculator to find the measure of θ.

17. 11. 4 4 9 9 θ = cosθ adj 63.6° cos–1 = = hyp Find the measure of the angle θ. SOLUTION In the right triangle, you are given the lengths of the side adjacent to θand the hypotenuse. So, use the inverse cosine function to solve for θ.

18. 12. 10 10 8 8 θ tan–1 = tanθ = opp 51.3° = adj Find the measure of the angle θ. SOLUTION In the right triangle, you are given the lengths of the side opposite to θand the side adjacent. So, use the inverse tan function to solve for θ.

19. 13. 5 5 12 12 θ sin–1 = sinθ = opp 24.6° = hyp Find the measure of the angle θ. SOLUTION In the right triangle, you are given the lengths of the side opposite to θand the hypotenuse. So, use the inverse sin function to solve for θ.

20. 9.4 Assignment Page 582, 3-29 odd