Evaluate Inverse Trigonometric Functions. 5p, }} 13p, }}

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13.4 a.1, a.3, 2A.4.C; P.3.A TEKS Evalate Inverse Trigonometri Fntions Before Yo fond vales of trigonometri fntions given angles.

Ke Voablar inverse sine inverse osine inverse tangent So far in this hapter, o have learned to evalate trigonometri fntions of a given angle.

After onsidering the problem, o wold realize man sh angles eist. For instane, the angles p }, }} 5p, }} 13p, }} 17p, and 2}} 7p 6 6 6 6 6 all have a sine vale of 0.5. To obtain a niqe angle sh that sin 5 0.

KEY CONCEPT For Yor Notebook Inverse Trigonometri Fntions If 21 a 1, then the inverse sine of a is an angle, written 5 sin 21 a, where: π 2 (1) sin 5 a (2) 2 p } 2 p } 2 (or

1 13.4 a.1, a.3, 2A.4.C; P.3.A TEKS Evalate Inverse Trigonometri Fntions Before Yo fond vales of trigonometri fntions given angles. Now Yo will find angles given vales of trigonometri fntions. Wh? So o an find lanh angles, as in Eample 4. Ke Voablar inverse sine inverse osine inverse tangent So far in this hapter, o have learned to evalate trigonometri fntions of a given angle. In this lesson, o will std the reverse problem finding an angle that orresponds to a given vale of a trigonometri fntion. Sppose o were asked to find an angle whose sine is 0.5. After onsidering the problem, o wold realize man sh angles eist. For instane, the angles p }, }} 5p, }} 13p, }} 17p, and 2}} 7p all have a sine vale of 0.5. To obtain a niqe angle sh that sin 5 0.5, o mst restrit the domain of the sine fntion. Domain restritions allow the inverse sine, inverse osine, and inverse tangent fntions to be defined. KEY CONCEPT For Yor Notebook Inverse Trigonometri Fntions If 21 a 1, then the inverse sine of a is an angle, written 5 sin 21 a, where: π 2 (1) sin 5 a (2) 2 p } 2 p } 2 (or ) 2 π 2 If 21 a 1, then the inverse osine of a is an angle, written 5 os 21 a, where: (1) os 5 a (2) 0 π (or ) π 0 If a is an real nmber, then the inverse tangent of a is an angle, written 5 tan 21 a, where: (1) tan 5 a π 2 (2) 2 p } 2 < < p } 2 (or 2908 < < 908) 2 π Evalate Inverse Trigonometri Fntions 875

2 E XAMPLE 1 Evalate inverse trigonometri fntions Evalate the epression in both radians and degrees. a. os 21 Ï} 3 b. sin tan 21 (2Ï } 3) Soltion a. When 0 π, or , the angle whose osine is Ï} 3 is: 5 os 21 Ï} 3 }} 5 } p or 5 os 21 Ï} 3 }} b. There is no angle whose sine is 2. So, sin 21 2 is ndefined.. When 2 p } 2 < < p } 2, or 2908 < < 908, the angle whose tangent is 2Ï } 3 is: 5 tan 21 (2Ï } 3) 52 p } 3 or 5 tan 21 (2Ï } 3) E XAMPLE 2 Solve a trigonometri eqation Solve the eqation sin 52 5 } 8 where 1808 < < Soltion USE A CALCULATOR On most allators, o an evalate inverse trigonometri fntions sing the kes for inverse sine, for inverse osine, and for inverse tangent. STEP 1 STEP 2 Use a allator to determine that in the interval , the angle whose sine is 2} 5 is sin } ø This angle is in Qadrant IV, as shown. Find the angle in Qadrant III (where 1808 < < 2708) that has the same sine vale as the angle in Step 1. The angle is: ø CHECK Use a allator to hek the answer. sin ø } 8 GUIDED PRACTICE for Eamples 1 and 2 Evalate the epression in both radians and degrees. 1. sin 21 Ï} 2 2. os 21 1 } 2 3. tan 21 (21) 4. sin } 2 2 Solve the eqation for. 5. os 5 0.4; 2708 < < tan 5 2.1; 1808 < < sin ; 2708 < < tan 5 4.7; 1808 < < sin ; 908 < < os ; 1808 < < Chapter 13 Trigonometri Ratios and Fntions

E XAMPLE 3 TAKS PRACTICE: Mltiple Choie What is the measre of the angle in the triangle shown? A 28.68 B 33.18 C 56.98 D 61.48 5 14 AVOID ERRORS All the answer hoies are in degrees.

Soltion In the right triangle, o are given the lengths of the side adjaent to and the hpotense, so se the inverse osine fntion to solve for. adj 5 os 5 }} 5 }} 5 os 21 }} 5 ø 69.

The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle of the ramp? Soltion STEP 1 STEP 2 Draw a triangle that represents the ramp.

3 E XAMPLE 3 TAKS PRACTICE: Mltiple Choie What is the measre of the angle in the triangle shown? A B C D AVOID ERRORS All the answer hoies are in degrees. Therefore, hek that or allator is set in degree mode, not radian mode. Soltion In the right triangle, o are given the lengths of the side adjaent to and the hpotense, so se the inverse osine fntion to solve for. adj 5 os 5 }} 5 }} 5 os 21 }} 5 ø hp The orret answer is C. A B C D E XAMPLE 4 Write and solve a trigonometri eqation MONSTER TRUCKS A monster trk drives off a ramp in order to jmp onto a row of ars. The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle of the ramp? Soltion STEP 1 STEP 2 Draw a triangle that represents the ramp. Write a trigonometri eqation that involves the ratio of the ramp s height and horizontal length. tan 5 }} opp 5 }} 8 adj 20 STEP 3 Use a allator to find the measre of. 5 tan ø The angle of the ramp is abot ft 8 ft GUIDED PRACTICE for Eamples 3 and 4 Find the measre of the angle WHAT IF? In Eample 4, sppose a monster trk drives 26 feet on a ramp before jmping onto a row of ars. If the ramp is 10 feet high, what is the angle of the ramp? 13.4 Evalate Inverse Trigonometri Fntions 877

13.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 7, 23, and 37 5 TAKS PRACTICE AND REASONING Es. 11, 30, 31, 37, 38, 41, and 42 1. VOCABULARY Cop and omplete: The?

4 13.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 7, 23, and 37 5 TAKS PRACTICE AND REASONING Es. 11, 30, 31, 37, 38, 41, and VOCABULARY Cop and omplete: The? sine of 1 } 2 is p } 6, or WRITING Eplain wh tan 21 3 is defined, bt os 21 3 is ndefined. EXAMPLE 1 on p. 876 for Es EVALUATING EXPRESSIONS Evalate the epression withot sing a allator. Give or answer in both radians and degrees. 3. sin tan 21 (21) 5. os os 21 (22) 7. sin 21 Ï} 3 }} 8. sin 21 } 1 9. tan Ï} 3 }} os } What is the vale of the epression os 21 Ï} 2 TAKS REASONING }}? 2 A 08 B 308 C 458 D 608 USING A CALCULATOR Use a allator to evalate the epression in both radians and degrees. 12. sin tan os os 21 (20.4) 16. tan 21 (20.75) 17. sin 21 (20.2) 18. sin os EXAMPLE 2 on p. 876 for Es SOLVING EQUATIONS Solve the eqation for. 20. os ; 1808 < < sin ; 1808 < < sin ; 908 < < tan 5 3.2; 1808 < < tan 525.3; 908 < < os ; 2708 < < ERROR ANALYSIS Desribe and orret the error in solving the eqation sin where 908 < < The angle whose sine is 0.7 is sin ø 44.48, so ø EXAMPLE 3 on p. 877 for Es FINDING ANGLES Find the measre of the angle TAKS REASONING Sppose os > 0 and sin < 0. Give a possible vale of sh that TAKS REASONING Sppose sin < 0 and tan > 0. Give a possible vale of sh that CHALLENGE Rewrite the epression so that it does not involve trigonometri fntions or inverse trigonometri fntions. 32. s (sin 21 ) 33. ot (tan 21 ) 34. se (os 21 ) 878 Chapter 13 Trigonometri Ratios and Fntions

PROBLEM SOLVING EXAMPLE 4 on p. 877 for Es. 35 37 35. LADDER ANGLE A fire trk has a 100 foot ladder whose base is 10 feet above the grond.

ANGLE OF DESCENT An airplane is fling at an altitde of 31,000 feet when it begins its desent for landing. If the rnwa is 104 miles awa, at what angle does the airplane desend? 37.

When rok salt is stored in a one-shaped pile 11 feet high, the diameter of the pile s base is abot 34 feet. Find the angle of repose for rok salt.

5 PROBLEM SOLVING EXAMPLE 4 on p. 877 for Es LADDER ANGLE A fire trk has a 100 foot ladder whose base is 10 feet above the grond. A firefighter etends a ladder toward a brning bilding to reah a window 90 feet above the grond. Draw a diagram to represent this sitation. At what angle shold the firefighter set the ladder? 36. ANGLE OF DESCENT An airplane is fling at an altitde of 31,000 feet when it begins its desent for landing. If the rnwa is 104 miles awa, at what angle does the airplane desend? 37. TAKS REASONING Different tpes of granlar sbstanes natrall settle at different angles when stored in one-shaped piles. The angle is alled the angle of repose. When rok salt is stored in a one-shaped pile 11 feet high, the diameter of the pile s base is abot 34 feet. Find the angle of repose for rok salt. If another pile of rok salt is 15 feet high, what is the diameter of its base? Eplain. 38. TAKS REASONING If o are in shallow water and look at an objet below the srfae of the water, the objet will look farther awa from o than it reall is. This is bease when light ras pass between air and water, the water refrats, or bends, the light ras. The inde of refration for water is This is the ratio of the sine of 1 to the sine of 2 for the angles 1 and 2 shown below. a. Yo are in 4 feet of water in the shallow end of a pool. Yo look down at some goggles at angle (measred from a line perpendilar to the srfae of the water). Find 2. b. Find the distanes and.. Find the distane d between where the goggles are and where the appear to be. d. Eplain what happens to d as o move loser to the goggles. 39. CYCLING As a spetator at a ling road rae, o are sitting 100 feet from the enter of a straightawa. A list traveling 30 miles per hor passes in front of o. At what angle do o have to trn or head to see the list t seonds later? Assme the list is still on the straightawa and is traveling at a onstant speed. (Hint: First onvert 30 miles per hor to a speed v in feet per seond. The epression vt represents the distane, in feet, traveled b the list.) 13.4 Evalate Inverse Trigonometri Fntions 879

40. CHALLENGE Yo want to photograph a painting with a amera monted on a tripod. The painting is 3 feet tall, and the bottom of the painting is 1 foot above the amera lens, as shown.

(Hint: Write an eqation for in terms of onl, and then se a graphing allator to find the vale of that maimizes.) MIXED REVIEW FOR TAKS TAKS PRACTICE at lasszone.om REVIEW Lesson 2.

6 40. CHALLENGE Yo want to photograph a painting with a amera monted on a tripod. The painting is 3 feet tall, and the bottom of the painting is 1 foot above the amera lens, as shown. How far shold the amera be positioned from the wall in order to have the largest possible viewing angle when o take the photograph? (Hint: Write an eqation for in terms of onl, and then se a graphing allator to find the vale of that maimizes.) MIXED REVIEW FOR TAKS TAKS PRACTICE at lasszone.om REVIEW Lesson 2.3; TAKS Workbook REVIEW Skills Review Handbook p. 992; TAKS Workbook 41. TAKS PRACTICE The graph of whih linear eqation has a slope of 2 2 } 5? TAKS Obj. 3 A B C D TAKS PRACTICE Steve plants a flower bed net to a orner of a bilding. The flower bed forms part of a irle with a radis of 10 feet. What is the flower bed s approimate area? TAKS Obj. 8 F 47.1 ft 2 G 78.5 ft 2 H ft 2 J ft 2 10 ft QUIZ for Lessons Use the given point on the terminal side of an angle in standard position to evalate the si trigonometri fntions of. (p. 866) 1. (6,22) 2. (27, 5) 3. (4, 8) 4. (212, 23) Evalate the epression withot sing a allator. (p. 866) 5. os tan }} 8p 7. sin (28408) 8. se p }} 4 2 Evalate the epression withot sing a allator. Give or answer in both radians and degrees. (p. 875) 9. os Ï} sin 21 (21) 11. tan 21 Ï} 3 }} 12. os 21 } Solve the eqation for. (p. 875) 13. sin 5 0.3; 908 < < tan 5 6; 1808 < < os ; 908 < < sin ; 2708 < < ACROBATICS A stntman ses a 30 foot rope to swing 1368 between two platforms of eqal height, grazing the grond in the middle of the swing. If the rope stas tat throghot the swing, how far above the grond was the stntman at the beginning and the end of the swing? How far apart are the two platforms? (p. 875) 880 EXTRA PRACTICE for Lesson 13.4, p ONLINE QUIZ at lasszone.om

Geometr Software ACTIVITY Use before Lesson 13.5 13.

om Kestrokes QUESTION How an o se geometr software to eplore the law of

E XPLORE Investigate a relationship between the angles and sides of a

Also find the measres of angles A, B, and C. C b A a B a 5 3.25 b 5 2.

28 C b A a B STEP 3 Callate ratios Find the ratios sin A } a, sin B } b,

28 DRAW CONCLUSIONS Use or observations to omplete these eerises 1.

7 Geometr Software ACTIVITY Use before Lesson Eplore the Law of Sines TEKS TEXAS a.4, a.5, a.6; P.3.E lasszone.om Kestrokes QUESTION How an o se geometr software to eplore the law of sines? E XPLORE Investigate a relationship between the angles and sides of a triangle STEP 1 Draw a triangle Draw n ABC. Label the verties and sides as shown. STEP 2 Measre parts of triangle Find the side lengths a, b, and. Also find the measres of angles A, B, and C. C b A a B a b aa ab ac C b A a B STEP 3 Callate ratios Find the ratios sin A } a, sin B } b, and sin C }. C a sin A b a A B sin B b sin C DRAW CONCLUSIONS Use or observations to omplete these eerises 1. What are the vales of the ratios} sin A, } sin B, and} sin C for or triangle? a b What do o notie abot these vales? 2. Change the shape of or triangle b dragging its verties, and observe how the ratios o fond in Step 3 hange. Make a onjetre abot how these ratios are related for an triangle Appl the Law of Sines 881

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 ).

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