1-4 Inverses of Trigonometric Functions Objectives Evaluate inverse trigonometric functions. Use trigonometric equations and inverse trigonometric functions to solve problems. Vocabulary inverse sine function inverse cosine function inverse tangent function Who uses this? Hikers can use inverse trigonometric functions to navigate in the wilderness. (See Example.) You have evaluated trigonometric functions for a given angle. You can also find the measure of angles given the value of a trigonometric function by using an inverse trigonometric relation. Function sin θ = a Inverse Relation si n a = θ cos θ = a co s a = θ tan θ = a ta n a = θ The expression si n is read as the inverse sine. In this notation, indicates the inverse of the sine function, NOT the reciprocal of the sine function. The inverses of the trigonometric functions are not functions themselves because there are many values of θ for a particular value of a. For example, suppose that you want to find cos 1. Based on the unit circle, angles that measure π and 5π radians have a cosine of 1. So do all angles that are coterminal with these angles. EXAMPLE 1 Finding Trigonometric Inverses California Standards Preview of Trigonometry 8.0 Students know the definitions of the inverse trigonometric functions and can graph the functions. Also covered: Preview of Trig 19.0 Find all possible values of si n. Step 1 Find the values between 0 and π radians for which sin θ is equal to. _ = sin _ π 4, _ = sin _ π 4 Step Find the angles that are coterminal with angles measuring π and π 4 4 radians. Use y-coordinates of points on the unit circle. π_ 4 + (π) n, π_ 4 + (π) n Add integer multiples of π radians, where n is an integer. 1. Find all possible values of ta n 1. Because more than one value of θ produces the same output value for a given trigonometric function, it is necessary to restrict the domain of each trigonometric function in order to define the inverse trigonometric functions. 950 Chapter 1 Trigonometric Functions
Trigonometric functions with restricted domains are indicated with a capital letter. The domains of the Sine, Cosine, and Tangent functions are restricted as follows. Sin θ = sin θ for θ - π_ θ _ π θ is restricted to Quadrants I and IV. Cos θ = cos θ for θ 0 θ π θ is restricted to Quadrants I and II. Tan θ = tan θ for θ - π_ θ is restricted to Quadrants I and IV. < θ < _ π These functions can be used to define the inverse trigonometric functions. For each value of a in the domain of the inverse trigonometric functions, there is only one value of θ. Therefore, even though ta n 1 has many values, Tan 1 has only one value. Inverse Trigonometric Functions The inverse trigonometric functions are also called the arcsine, arccosine, and arctangent functions. WORDS SYMBOL DOMAIN RANGE - π_ The inverse sine function is Si n a = θ, where Sin θ = a. The inverse cosine function is Co s a = θ, where Cos θ = a. The inverse tangent function is Ta n a = θ, where Tan θ = a. Si n a a a 1 Co s a a a 1 Ta n a a - < a < θ _ π θ -90 θ 90 θ 0 θ π θ 0 θ 180 θ - π_ < θ < _ π θ -90 < θ < 90 EXAMPLE Evaluating Inverse Trigonometric Functions Evaluate each inverse trigonometric function. Give your answer in both radians and degrees. A Cos 1_ 1_ = Cos θ Find the value of θ for 0 θ π whose Cosine is _ 1. 1_ π_ = Cos Use x-coordinates of points on the unit circle. Cos 1 _ = π _, or Co s 1 _ = 60 B Sin The domain of the inverse sine function is a a 1. Because is outside this domain, Si n is undefined. Evaluate each inverse trigonometric function. Give your answer in both radians and degrees. a. Si n _ ) b. Co s 0 1-4 Inverses of Trigonometric Functions 951
You can solve trigonometric equations by using trigonometric inverses. EXAMPLE Navigation Application A group of hikers plans to walk from a campground to a lake. The lake is miles east and 0.5 mile north of the campground. To the nearest degree, in what direction should the hikers head? Step 1 Draw a diagram. The hikers direction should be based on θ, the measure of an acute angle of a right triangle. If the answer on your calculator screen is 0.44978661 when you enter ta n (0.5), your calculator is set to radian mode instead of degree mode. Step Find the value of θ. tan θ = _ opp. adj. Use the tangent ratio. tan θ = _ 0.5 = 0.5 Substitute 0.5 for opp. and for adj. Then simplify. θ = Tan 0.5 θ 14 The hikers should head 14 north of east. Use the information given above to answer the following.. An unusual rock formation is 1 mile east and 0.75 mile north of the lake. To the nearest degree, in what direction should the hikers head from the lake to reach the rock formation? EXAMPLE 4 Solving Trigonometric Equations Solve each equation to the nearest tenth. Use the given restrictions. A cos θ = 0.6, for 0 θ 180 The restrictions on θ are the same as those for the inverse cosine function. θ = Co s (0.6) 5.1 Use the inverse cosine function on your calculator. B cos θ = 0.6, for 70 < θ < 60 The terminal side of θ is restricted to Quadrant IV. Find the angle in Quadrant IV that has the same cosine value as 5.1. θ 60-5.1 06.9 θ has a reference angle of 5.1, and 70 < θ < 60. Solve each equation to the nearest tenth. Use the given restrictions. 4a. tan θ = -, for -90 < θ < 90 4b. tan θ = -, for 90 < θ < 180 95 Chapter 1 Trigonometric Functions
THINK AND DISCUSS 1. Given that θ is an acute angle in a right triangle, describe the measurements that you need to know to find the value of θ by using the inverse cosine function.. Explain the difference between tan a and Tan a.. GET ORGANIZED Copy and complete the graphic organizer. In each box, give the indicated property of the inverse trigonometric functions. 1-4 Exercises California Standards Preview of Trig 8.0 and 19.0; 4.0 KEYWORD: MB7 1-4 GUIDED PRACTICE KEYWORD: MB7 Parent 1. Vocabulary Explain how the inverse tangent function differs from the reciprocal of the tangent function. SEE EXAMPLE 1 p. 950 SEE EXAMPLE p. 951 SEE EXAMPLE p. 95 Find all possible values of each expression.. sin ( - 1_. tan _ ) 4. co s _ ) Evaluate each inverse trigonometric function. Give your answer in both radians and degrees. 5. Cos _ 6. Tan 1 7. Cos 8. Tan ) 9. Sin _ 10. Sin 0 11. Architecture A point on the top of the Leaning Tower of Pisa is shifted about 1.5 ft horizontally compared with the tower s base. To the nearest degree, how many degrees does the tower tilt from vertical? SEE EXAMPLE 4 p. 95 Solve each equation to the nearest tenth. Use the given restrictions. 1. tan θ = 1.4, for -90 < θ < 90 1. tan θ = 1.4, for 180 < θ < 70 14. cos θ = -0.5, for 0 θ 180 15. cos θ = -0.5, for 180 < θ < 70 1-4 Inverses of Trigonometric Functions 95
Independent Practice For See Exercises Example 16 18 1 19 4 5 6 9 4 Extra Practice Skills Practice p. S9 Application Practice p. S44 Aviation A flight simulator is a device used in training pilots that mimics flight conditions as realistically as possible. Some flight simulators involve fullsize cockpits equipped with sound, visual, and motion systems. PRACTICE AND PROBLEM SOLVING Find all possible values of each expression. 16. cos 1 17. sin _ 18. tan () Evaluate each inverse trigonometric function. Give your answer in both radians and degrees. 19. Sin _ 0. Cos () 1. Tan _ ). Cos _ ). Tan 4. Sin 5. Volleyball A volleyball player spikes the ball from a height of.44 m. Assume that the path of the ball is a straight line. To the nearest degree, what is the maximum angle θ at which the ball can be hit and land within the court? Solve each equation to the nearest tenth. Use the given restrictions. 6. sin θ = -0.75, for -90 θ 90 7. sin θ = -0.75, for 180 < θ < 70 8. cos θ = 0.1, for 0 θ 180 9. cos θ = 0.1, for 70 < θ < 60 0. Aviation The pilot of a small plane is flying at an altitude of 000 ft. The pilot plans to start the final descent toward a runway when the horizontal distance between the plane and the runway is mi. To the nearest degree, what will be the angle of depression θ from the plane to the runway at this point? 1. Multi-Step The table shows the dimensions of three pool styles offered by a construction company. a. To the nearest tenth of a degree, what angle θ does the bottom of each pool make with the horizontal? b. Which pool style s bottom has the steepest slope? Explain. c. What if...? If the slope of the bottom of a pool can be no greater than 1, what is the greatest 6 angle θ that the bottom of the pool can make with the horizontal? Round to the nearest tenth of a degree. Pool Style Length (ft) Shallow End Depth (ft) Deep End Depth (ft) A 8 8 B 5 6 C 50.5 7. Navigation Lines of longitude are closer together near the poles than at the equator. The formula for the length l of 1 of longitude in miles is l = 69.09 cos θ, where θ is the latitude in degrees. a. At what latitude, to the nearest degree, is the length of a degree of longitude approximately 59.8 miles? b. To the nearest mile, how much longer is the length of a degree of longitude at the equator, which has a latitude of 0, than at the Arctic Circle, which has a latitude of about 66 N? 954 Chapter 1 Trigonometric Functions
. This problem will prepare you for the Concept Connection on page 956. Giant kelp is a seaweed that typically grows about 100 ft in height, but may reach as high as 175 ft. a. A diver positions herself 10 ft from the base of a giant kelp so that her eye level is 5 ft above the ocean floor. If the kelp is 100 ft in height, what would be the angle of elevation from the diver to the top of the kelp? Round to the nearest tenth of a degree. b. The angle of elevation from the diver s eye level to the top of a giant kelp whose base is 0 ft away is 75.5. To the nearest foot, what is the height of the kelp? Find each value. 4. Co s (cos 0.4) 5. tan (Tan 0.7) 6. sin (Cos 0) 7. Critical Thinking Explain why the domain of the Cosine function is different from the domain of the Sine function. 8. Write About It Is the statement Sin (sin θ) = θ true for all values of θ? Explain. 9. For which equation is the value of θ in radians a positive value? Cos θ = -_ 1 Tan θ = -_ Sin θ = -_ Sin θ = 40. A caution sign next to a roadway states that an upcoming hill has an 8% slope. An 8% slope means that there is an 8 ft rise for 100 ft of horizontal distance. At approximately what angle does the roadway rise from the horizontal?. 4.6 8.5 1.5 41. What value of θ makes the equation (Cos θ) = - true? 45 60 15 150 CHALLENGE AND EXTEND 4. If Sin _ ) = -_ π 4, what is the value of Csc )? Solve each inequality for θ 0 θ π. 4. cos θ _ 1 44. sin θ - > 0 45. tan θ 1 SPIRAL REVIEW Graph each function. Identify the parent function that best describes the set of points, and describe the transformation from the parent function. (Lesson 1-9) 46., -4), (, -0.5), (0, 0), (1, 0.5), (, 4) 47. 4, 1),, ), (0, 5), (, 7), (4, 9) Find the inverse of each function. Determine whether the inverse is a function, and state its domain and range. (Lesson 9-5) 48. f (x) = (x + ) 49. f (x) = _ x 4 + 1 50. f (x) = - x + 5 Convert each measure from degrees to radians or from radians to degrees. (Lesson 1-) 51. 40 5. - 5π_ 5. 40 4 1-4 Inverses of Trigonometric Functions 955
Quiz for Lessons 1 Through 1-4 SECTION 1A 1 Right-Angle Trigonometry Find the values of the six trigonometric functions for θ. 1.. Use a trigonometric function to find the value of x.. 4. 5. A biologist s eye level is 5.5 ft above the ground. She measures the angle of elevation to an eagle s nest on a cliff to be 66 when she stands 50 ft from the cliff s base. To the nearest foot, what is the height of the eagle s nest? 1- Angles of Rotation Draw an angle with the given measure in standard position. 6. -70 7. 405 Point P is a point on the terminal side of θ in standard position. Find the exact value of the six trigonometric functions for θ. 8. P (1, -5) 9. P, 7) 1- The Unit Circle Convert each measure from degrees to radians or from radians to degrees. 10. 0 11. 6 1. π_ 1. -_ 10π 8 Use the unit circle to find the exact value of each trigonometric function. 14. cos 10 15. tan 10 16. cos _ π 17. tan 5π _ 4 18. A bicycle tire rotates through an angle of.4π radians in 1 second. If the radius of the tire is 0.4 m, what is the bicycle s speed in meters per second? Round to the nearest tenth. 1-4 Inverses of Trigonometric Functions Evaluate each inverse trigonometric function. Give your answer in both radians and degrees. 19. Si n _ 0. Ta n _ ) 1. A driver uses a ramp when unloading supplies from his delivery truck. The ramp is 10 feet long, and the bed of the truck is 4 feet off the ground. To the nearest degree, what angle does the ramp make with the ground? Ready to Go On? 957