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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? Inverse Trig Functions y x Inverse Trig Functions y 0 x Inverse Trig Functions y x Evaluate the expression in both radians and degrees. a. cos–1 √ 3 2 SOLUTION a. When 0 ≤ θ ≤ π or 0°≤ θ ≤ whose cosine is √ 3 2 π √3 –1 cos = θ = 2 6 180°, the angle √ 3 30° = θ = cos–1 2 y 90° 120° 60° 45° 135° 30° 150° 0° 360° 180° 330° 210° 315° 225° 240° 300° 270° x Evaluate the expression in both radians and degrees. b. sin–1 2 SOLUTION b. There is no angle whose sine is 2. So, sin–1 2 is undefined. Evaluate the expression in both radians and degrees. c. tan–1 ( – √ 3 ) SOLUTION c. When – π < θ < π , or – 90° < θ < 90°, the 2 2 angle whose tangent is – √ 3 is: θ= tan–1 π – (–√3 ) = 3 θ = tan–1 ( – √ 3 ) = –60° Evaluate the expression in both radians and degrees. 1. sin–1 √ 2 2 ANSWER 2. 3. π , 45° 4 cos–1 1 2 ANSWER ANSWER – 4π , –45° 4. π , 60° 3 tan–1 (–1) sin–1 (– 1 ) 2 ANSWER – 6π , –30° Solve a Trigonometric Equation 5 Solve the equation sin θ = – 8 where 180° < θ < 270°. SOLUTION STEP 1 Use a calculator to determine that in the interval –90° ≤ θ ≤ 90°, the angle whose 5 sine is – 5 is sin–1 – – 38.7°. This 8 8 angle is in Quadrant IV, as shown. 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 – sin 218.7° – 0.625 = 8 Solve the equation for 5. cos θ = 0.4; ANSWER 6. tan θ = 2.1; ANSWER 7. sin θ = –0.23; ANSWER 270° < θ < 360° about 293.6° 360 66.4 293.6 180° < θ < 270° about 244.5° 64.5 180 244.5 270° < θ < 360° about 346.7° 360 13.3 346.7 Solve the equation for 8. tan θ = 4.7; ANSWER 9. sin θ = 0.62; ANSWER 180° < θ < 270° about 258.0° 78 180 258 90° < θ < 180° about 141.7° 180 38.3 141.7 10. cos θ = –0.39; 180° < θ < 270° ANSWER about 247.0° 360 113 247 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 = 6 adj = 11 hyp θ = cos ANSWER The correct answer is C. –1 6 11 56.9° 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 SOLUTION STEP 1 Draw: a triangle that represents the ramp. Write: a trigonometric equation STEP 2 that involves the ratio of the ramp’s height and horizontal length. 8 opp tan θ = = 20 adj STEP 3 Use: a calculator to find the measure of θ. θ= tan–1 8 20 21.8° ANSWER The angle of the ramp is about 22°. 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 4 = 9 hyp θ = cos–1 4 9 63.6° Find the measure of the angle θ. 12. 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. tan θ = opp 10 = 8 adj θ = tan–1 10 8 51.3° Find the measure of the angle θ. 13. 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. sin θ = opp 5 = 12 hyp θ = sin–1 5 12 24.6° 9.4 Assignment Page 582, 3-29 odd