1. 9.3 Use Trig Functions to Find the Measure of the Angle

2. N EW V OCABULARY :

4. What measurement of angles are known? How can we find the other two angles? What kind of angles are they? Identify each side of the triangle using hypotenuse, opposite, and adjacent

5. L ABILE EACH SIDE OF THE T RIANGLE Hypotenuse “H” Opposite “O” Adjacent “A”

6. EXAMPLE 1 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

7. You Try: for Examples 1 Find the measure of the angle θ. 1. 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° θ cos –1 4 9

8. You Try: for Examples 1 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. 2. tan θ = opp adj = 10 8 θ 51.3° = tan –1 10 8

9. You Try: for Examples 1 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. 3. sin θ = opp hyp = 5 12 24.6° θ = sin –1 5 12

10. EXAMPLE 2 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?

11. EXAMPLE 2 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

12. EXAMPLE 2 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

13. You Try for Examples 2 4. WHAT IF? In Example 2, 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

14. You try: for Examples 2 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