Inverse Trigonometric Functions. inverse sine, inverse cosine, and inverse tangent are given below. where tan = a and º π 2 < < π 2 (or º90 < < 90 ).

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1 Page 1 of 7 1. Inverse Trigonometric Functions What ou should learn GOAL 1 Evaluate inverse trigonometric functions. GOAL Use inverse trigonometric functions to solve real-life problems, such as finding an angle of repose in Eample. Wh ou should learn it To solve real-life problems, such as finding the angle at which to set the arm of a crane in Eample 5. GOAL 1 EVALUATING AN INVERSE TRIGONOMETRIC FUNCTION In the first three lessons of this chapter, ou learned to evaluate trigonometric functions of a given angle. In this lesson ou will stud the reverse problem finding angles that correspond to a given value of a trigonometric function. Suppose ou were asked to find an angle whose sine is 0.5. After thinking about the problem for a while, ou would probabl realize that there are man such angles. For instance, the angles π, 5 π 1π,, 17 π, and º 7 π all have a sine value of 0.5. (Tr checking this with a calculator.) Of these, the value of the inverse sine function at 0.5 is defined to be π. General definitions of inverse sine, inverse cosine, and inverse tangent are given below. INVERSE TRIGONOMETRIC FUNCTIONS If º1 a 1, then the inverse sine of a is sin º1 a = where sin = a and º π π (or º0 0 ). º π π º1 sin 1 If º1 a 1, then the inverse cosine of a is cos º1 a = where cos = a and 0 π (or ). 0 π º1 cos 1 If a is an real number, then the inverse tangent of a is tan º1 a = where tan = a and º π < < π (or º0 < < 0 ). º π < < π º < tan < + 7 Chapter 1 Trigonometric Ratios and Functions

2 Page of 7 EXAMPLE 1 Evaluating Inverse Trigonometric Functions Evaluate the epression in both radians and degrees. a. sin º1 b. cos º1 c. tan º1 (º1) a. When º π π, or º0 0, the angle whose sine is is: = sin º1 = π or = sinº1 = 0 b. There is no angle whose cosine is. So, cos º1 is undefined. c. When º π < < π, or º0 < < 0, the angle whose tangent is º1 is: = tan º1 (º1) = º π or = tanº1 (º1) = º5 EXAMPLE Finding an Angle Measure Stud Tip When approimating the value of an angle, make sure our calculator is set to radian mode if ou want our answer in radians, or to degree mode if ou want our answer in degrees. Find the measure of the angle for the triangle shown. In the right triangle, ou are given the adjacent side and the hpotenuse. You can write: adj cos = = 5 h p This equation is asking ou to find the acute angle whose cosine is 5. Use a calculator to find the measure of. = cos º radians or = cosº EXAMPLE Solving a Trigonometric Equation Solve the equation sin = º 1 where 180 < < 70. In the interval º0 < < 0, the angle whose sine is º 1 is sinº1 º 1 º1.5. This angle is in Quadrant IV as shown. In Quadrant III (where 180 < < 70 ), the angle that has the same sine value is: = CHECK Use a calculator to check the answer. sin 1.5 º Inverse Trigonometric Functions 7

3 Page of 7 FOCUS ON APPLICATIONS GOAL USING INVERSE TRIGONOMETRIC FUNCTIONS IN EXAMPLE Writing and Solving a Trigonometric Equation ROCK SALT Each ear about million tons of rock salt are poured on highwas in North America to melt ice. Although rock salt is the best deicing material, it also eats awa at cars and road surfaces. ROCK SALT Different tpes of granular substances naturall settle at different angles when stored in cone-shaped piles. This angle is called the angle of repose. When rock salt is stored in a cone-shaped pile 11 feet high, the diameter of the pile s base is about feet. Source: Bulk-Store Structures, Inc. a. Find the angle of repose for rock salt. b. How tall is a pile of rock salt that has a base diameter of 50 feet? a. In the right triangle shown inside the cone, ou are given the opposite side and the adjacent side. You can write: tan = o pp 11 = adj 1 7 This equation is asking ou to find the acute angle whose tangent is = tan º The angle of repose for rock salt is about.. θ 17 ft 11 ft b. The pile of rock salt has a base radius of 5 feet. From part (a) ou know that the angle of repose for rock salt is about.. To find the height h (in feet) of the pile ou can write: h tan. = 5 h = 5 tan. 1. The pile of rock salt is about 1. feet tall. EXAMPLE 5 Writing and Solving a Trigonometric Equation Construction A crane has a 00 foot arm whose lower end is 5 feet off the ground. The arm has to reach the top of a building 10 feet high. At what angle should the arm be set? In the right triangle in the diagram, ou know the opposite side and the hpotenuse. You can write: sin = o pp = 1 0 º 5 5 = hp ft This equation is asking ou to find the acute angle whose sine is 5 8. = sin º The crane s arm should be set at an acute angle of about ft 10 ft Not drawn to scale 7 Chapter 1 Trigonometric Ratios and Functions

4 Page of 7 GUIDED PRACTICE Vocabular Check Concept Check Skill Check 1. Complete this statement: The? sine of 1 equals π, or 0.. Eplain wh the domain of = cos cannot be restricted to º π π if the inverse is to be a function.. Eplain wh tan º1 is defined, but cos º1 is undefined.. ERROR ANALYSIS A student needed to find an angle in Quadrant III such that sin = º0.1. She used a calculator to find that sin º1 (º0.1) º18.8. Then she added this result to 180 to get an answer of = 11.. What did she do wrong? Evaluate the epression without using a calculator. 5. tan º1. cos º1 7. sin º cosº1 º 1 Use a calculator to evaluate the epression in both radians and degrees. Round to three significant digits.. tan º cos º1 (º0.) 11. cos º sin º1 (º0.) Solve the equation for. Round to three significant digits. 1. sin = º0.5; 180 < < tan =.; 180 < < cos = 0.; 70 < < 0 1. sin = 0.8; 0 < < CONSTRUCTION A crane has a 150 foot arm whose lower end is feet off the ground. The arm has to reach the top of a building 105 feet high. At what angle should the crane s arm be set? PRACTICE AND APPLICATIONS Etra Practice to help ou master skills is on p. 58. HOMEWORK HELP Eample 1: Es. 18 5, Eample : Es. Eample : Es. 51 Eamples, 5: Es EVALUATING EXPRESSIONS Evaluate the epression without using a calculator. Give our answer in both radians and degrees. 18. sin º1 1. cos º tanº sin º1 0. cos º1 (º1). sin º1 (º1). tan º1 º 5. cos º1 º FINDING ANGLES Find the measure of the angle. Round to three significant digits Inverse Trigonometric Functions 75

5 Page 5 of 7 EVALUATING EXPRESSIONS Use a calculator to evaluate the epression in both radians and degrees. Round to three significant digits.. tan º1.. cos º1 0.. cos º sin º sin º1 (º0.) 7. cos º1 (º0.) 8. tan º1 (º0.). tan º cos º1 (º0.8) 1. sin º1 0...tan º1 1. cos º HOMEWORK HELP Visit our Web site for help with Es. 51. INTERNET SOLVING EQUATIONS Solve the equation for. Round to three significant digits.. sin = º0.5; 180 < < tan =.; 180 < < 70. cos = 0.; 70 < < 0 7. sin = 0.8; 0 < < tan = º.1; 0 < < 180. cos = º0.7; 180 < < sin = 0.; 0 < < tan = 0.; 180 < < SWIMMING POOL The swimming pool shown in cross section at the right ranges in depth from feet at the shallow end to 8 feet at the deep end. Find the angle of depression between the shallow end and the deep end. 5. DUMP TRUCK The dump truck shown has a 10 foot bed. When tilted at its maimum angle, the bed reaches a height of 7 feet above its original position. What is the maimum angle that the truck bed can tilt? 5. GRANULAR ANGLE OF REPOSE Look back at Eample on page 7. When whole corn is stored in a cone-shaped pile 0 feet high, the diameter of the pile s base is about 8 feet. Find the angle of repose for whole corn. 55. ROAD DESIGN Curves that connect two straight sections of a road are often constructed as arcs of circles. In the diagram, is the central angle of a circular arc that has a radius of 5 feet. Each radius line shown is perpendicular to one of the straight sections. The straight sections are therefore tangent to the arc. The etension of each straight section to their point of intersection is 158 feet in length. Find the degree measure of. 5. DRAWBRIDGE The Park Street Bridge in Alameda Count, California, is a double-leaf drawbridge. Each leaf of the bridge is 10 feet long. A ship that is 100 feet wide needs to pass through the bridge. What is the minimum angle that each leaf of the bridge should be opened to in order to ensure that the ship will fit? Source: Alameda Count Drawbridges ft 10 ft 10 ft 5 ft 10 ft 0 ft 158 ft 100 ft 7 ft 5 ft 10 ft 158 ft 8 ft 10 ft 7 Chapter 1 Trigonometric Ratios and Functions

6 Page of 7 FOCUS ON PEOPLE 57. RACEWAY Suppose ou are at a racewa and are sitting on the straightawa, 100 feet from the center of the track. If a car traveling 15 miles per hour passes directl in front of ou, at what angle do ou have to turn our head to see the car t seconds later? Assume that the car is still on the straightawa and is traveling at a constant speed. (Hint: First convert 15 miles per hour to a speed v in feet per second. The epression vt represents the distance in feet traveled b the car.) vt 100 ft JEFF GORDON began racing in the top division of the National Association for Stock Car Auto Racing (NASCAR) in 1. He has won three NASCAR division championships in the past four ears. Test Preparation Challenge GEOMETRY CONNECTION In Eercises 58, use the following information. Consider a line with positive slope m that makes an angle with the -ais (measuring counterclockwise from the -ais). 58. Find the slope m of the line = º. 5. Find for the line = º. 0. CRITICAL THINKING How could ou have found for the line = º b using the slope m of the line? Write an equation relating and m. 1. Find an equation of the line that makes an angle of 58 with the -ais and whose -intercept is.. Find an equation of the line that makes an angle of 5 with the -ais and whose -intercept is.. MULTI-STEP PROBLEM If ou stand in shallow water and look at an object below the surface of the water, the object will look farther awa from ou than it reall is. This is because when light ras pass between air and water, the water refracts, or bends, the light ras. The inde of refraction for seawater is 1.1. This is the ratio of the sine of 1 to the sine of for angles 1 and below. a. You are standing in seawater that is feet deep and are looking at a shell at angle 1 =0 (measured from a line perpendicular to the surface of the water). Find. b. Find the distances and. c. Find the distance d between where the shell is and where it appears to be. d. Writing What happens to d as ou move closer to the shell? Eplain our reasoning. Not drawn to scale. LENGTH OF A PULLEY BELT Find the length of the pulle belt shown at the right. (Hint: Partition the belt into four parts: the two straight segments, the arc around the small wheel, and the arc around the large wheel.) 8 in. 1 ft 1 in. d in. 1. Inverse Trigonometric Functions 77

7 Page 7 of 7 MIXED REVIEW SOLVING EQUATIONS Solve the rational equation. Check for etraneous solutions. (Review. for 1.5) 5. = 7. = 7 + º 7. º 1 = + º =. = 70. = º º CHOOSING NUMBERS You have an equall likel chance of choosing an number 1 through 0. Find the probabilit of the given event. (Review 1.) 71. A multiple of 5 is chosen. 7. A prime number is chosen. 7. An even number is chosen. 7. A factor of 0 is chosen. 75. A number less than 1 is chosen. 7. A number greater than is chosen. EVALUATING FUNCTIONS Use a calculator to evaluate the function. Round to four decimal places. (Review 1. for 1.5) 77. sin sin π 7. cos sec 5 π 81. tan 1 8. csc 1 QUIZ Self-Test for Lessons 1. and 1. Use the given point on the terminal side of an angle in standard position. Evaluate the si trigonometric functions of. (Lesson 1.) 1. (º, º1). (7, º). (º1, 5). (, º11) 5. (, ). (º1, ) 7. (, º5) 8. (º7, º8) Evaluate the function without using a calculator. (Lesson 1.). sin (º15 ) 10. tan 8 π 11. cos (º0 ) 1. tan º π 1. sin 5 π π 1. cos tan (º0 ) 1. sin Use a calculator to evaluate the epression in both radians and degrees. Round to three significant digits. (Lesson 1.) 17. tan º sin º1 (º0.) 1. cos º sin º tan º1 (º). cos º1 (º0.8). sin º tan º1 10 Solve the equation for. Round to three significant digits. (Lesson 1.) 5. sin = 0.5; 0 < < 180. cos = 0.1; 70 < < 0 7. tan = 7; 180 < < sin = º0.; 180 < < 70. cos = º0.; 180 < < tan = º.5; 0 < < LACROSSE A lacrosse plaer throws a ball at an angle of 55 and at an initial speed of 0 feet per second. How far awa should her teammate be to catch the ball at the same height from which it was thrown? (Lesson 1.) 78 Chapter 1 Trigonometric Ratios and Functions