C H A P T E R 4 Trigonometric Functions

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1 C H A P T E R Trigonometric Functions Section. Radian and Degree Measure Section. Trigonometric Functions: The Unit Circle Section. Right Triangle Trigonometr Section. Trigonometric Functions of An Angle Section. Graphs of Sine and Cosine Functions Section. Graphs of Other Trigonometric Functions Section.7 Inverse Trigonometric Functions Section.8 Applications and Models Review Eercises Practice Test Houghton Mifflin Compan. All rights reserved.

2 C H A P T E R Trigonometric Functions Section. Radian and Degree Measure You should know the following basic facts about angles, their measurement, and their applications. Tpes of Angles: (a) Acute: Measure between 0 and 90. Right: Measure 90. (c) Obtuse: Measure between 90 and 80. (d) Straight: Measure 80. and are complementar if The are supplementar if Two angles in standard position that have the same terminal side are called coterminal angles. To convert degrees to radians, use 80 radians. To convert radians to degrees, use radian 80. one minute 0 of one second 0 of 00 of The length of a circular arc is s r where is measured in radians. Speed distancetime Angular speed t srt Vocabular Check. Trigonometr. angle. standard position. coterminal. radian. complementar 7. supplementar 8. degree 9. linear 0. angular. The angle shown is approimatel radians.. (a) Since lies in Quadrant IV. < 7 <, Since < <, Quadrant II. 7 lies in. The angle shown is approimatel radians.. (a) Since lies in < < 0, Quadrant IV. Since < <, 9 Quadrant II. 9 lies in Houghton Mifflin Compan. All rights reserved. 7

3 Section. Radian and Degree Measure 7. (a) Since < < 0; lies in Quadrant IV.. (a) Since <. <. lies in Quadrant III., Since < < lies in ; Quadrant III. Since. lies in Quadrant II. <. <, 7. (a) 8. (a) 7 π 7 π π π 9. (a) π 0. (a) Houghton Mifflin Compan. All rights reserved. π. (a) Coterminal angles for : Coterminal angles for : 8

4 7 Chapter Trigonometric Functions. (a) Coterminal angles for Coterminal angles for 7 :. (a) Coterminal angles for : Coterminal angles for 8 9 :. (a) Coterminal angles for : Coterminal angles for : : 8. Complement: Supplement: 7. Complement: Supplement:. Complement: Not possible; is greater than. (a)supplement: 8. Complement: Not possible; is greater than. Supplement: 9. Complement: Supplement: Complement: None >. Supplement:... The angle shown is approimatel 0.. (a) Since 90 < 0 < 80, 0 lies in Quadrant II. Since 70 < 8 < 0, 8 lies in Quadrant IV.. (a) Since 80 < 0 < 90, 0 lies in Quadrant III. Since 0 < 0 < 70, 0 lies in Quadrant I. The angle shown is approimatel.. (a) Since 0 < 87.9 < 90, 87.9 lies in Quadrant I. Since 0 < 8. < 90, 8. lies in Quadrant I.. (a) Since 70 <. < 80,. lies in Quadrant II. Since 90 <. < 0,. lies in Quadrant IV. Houghton Mifflin Compan. All rights reserved.

5 Section. Radian and Degree Measure 7 7. (a) (a) (a) 0 0. (a) Houghton Mifflin Compan. All rights reserved.. (a) Coterminal angles for : Coterminal angles for : (a) Coterminal angles for : Coterminal angles for 70: Complement: Supplement: (a) Coterminal angles for : Coterminal angles for 90: Complement: 90 Supplement: Complement: Not possible Supplement: (a) Coterminal angles for 00: Coterminal angles for 0: Complement: Not possible 9. (a) Supplement:

6 7 Chapter Trigonometric Functions 0. (a) (a) (a) (a). (a) (a). (a) radians 8. radians radians radians radians radians Houghton Mifflin Compan. All rights reserved.

7 Section. Radian and Degree Measure s r 7. s r 7. s r radians 7 radians radians 7. s r radians Because the angle represented is clockwise, this angle is radians. 7. The angles in radians are: The angles in degrees are: Houghton Mifflin Compan. All rights reserved. 77. s r 8 8 radians 80. r 80 kilometers, s 0 kilometers s r r 9 feet, radians 0 s r 9 feet s r radian s r, in radians s r. 70. radians 9 s 80 inches s r, in radians s 7 8 meters. meters

8 78 Chapter Trigonometric Functions 8. r centimeters, s r 8. r s 9 9 centimeters 8.7 cm meters 0.7 meter 8. r s 7 feet.9 feet 87. r s 8 8 miles.80 miles r s inches.9 inches radian s r miles 90. r 000 miles rad s r miles 9. s r radian r 78, s 00 s r rad.9 78 s r radian 9. s r.8 rad (a) single ael: double ael: revolutions radians revolutions radians (c) triple ael: revolutions 0 7 radians 9. Linear speed s t 97. (a) r t kmmin 0 Revolutions 00 Second 0 0 revsec 98. (a) Revolutions revsec Second 0 Angular speed 0 80 radsec Angular speed 80 0 radsec Radius of saw blade Radius in feet Speed s t.7 r r t 7..7 in. 0. ft rangular speed t ftsec Radius of saw blade Radius in feet Speed s t. r r t 7.. in. 0.0 ft rangular speed t ftsec Houghton Mifflin Compan. All rights reserved.

9 Section. Radian and Degree Measure (a) Revolutions Hour Angular speed 8,800 7,00 radhr Radius of wheel in.ft80 ftmi, miles Speed s t 80 8,800 revhr 0 r r t, 7,00.70 mileshr Let spin balance machine rate. 70 rangular speed r rangular speed t 0, revmin 00. (a) 00 radmin angular speed 000 radmin Speed s t 00 revolutions minute 00 angular speed 00 r r t 00 linear speed 000 For the outermost track, 000 cmmin 00 rangular speed angular speed t False, radian 7., so one radian is much larger than one degree. 0. No, 0 is coterminal with True: 8 80 Houghton Mifflin Compan. All rights reserved. 0. (a) An angle is in standard position when the origin is the verte and the initial side coincides with the positive -ais. A negative angle is generated b a clockwise rotation. (c) Angles that have the same initial and terminal sides are coterminal angles. (d) An obtuse angle is between 90 and If is constant, the length of the arc is proportional to the radius s r, and hence increasing. 07. A square meters r Let A be the area of a circular sector of radius r and central angle. Then A r 08. Because r. s r,. Hence, A r 90 ft.

10 80 Chapter Trigonometric Functions 09. A r, s r (a) 0.8 A r r Domain: r > 0 8 A s s r r0.8 Domain: r > 0 The area function changes more rapidl for r > because it is quadratic and the arc length function is linear. 0 0 r 0 A 0 0 s r 0 Domain: 0 < Domain: 0 < < < 0 A s If a fan of greater diameter is installed, the angular speed does not change.. Answers will var.. Answers will var Houghton Mifflin Compan. All rights reserved.

11 Section. Trigonometric Functions: The Unit Circle 8 Section. Trigonometric Functions: The Unit Circle You should know how to evaluate trigonometric functions using the unit circle. You should know the definition of a periodic function. You should be able to recognize even and odd trigonometric functions. You should be able to evaluate trigonometric functions with a calculator in both radian and degree mode. Vocabular Check. unit circle. periodic. odd, even. sin 7.,, cos 8 7 sin tan 8 cos cot 8 tan sec 7 8 csc 7 csc sec cot. sin.,, Houghton Mifflin Compan. All rights reserved. cos tan cot sec csc sin cos tan csc sec cot. t corresponds to,.. t,

12 8 Chapter Trigonometric Functions 7 7. t corresponds to,. 9. t corresponds to,. 8. t, 0. corresponds to t,.. t corresponds to 0,.. t, 0. corresponds to t 7,.. t corresponds to,.. t corresponds to 0,.. t corresponds to, t corresponds to sin t cos t tan t 7 9. t corresponds to sin t cos t tan t. t corresponds to sin t cos t tan t,.,.,. 8. t corresponds to sin cos tan,. 0. t corresponds to sin t cos t tan t. t corresponds to sin t cos t tan t,.,. Houghton Mifflin Compan. All rights reserved.

13 Section. Trigonometric Functions: The Unit Circle 8. t corresponds to sin t cos t tan t,.. t corresponds to sin t cos t tan t,.. t corresponds to sin t cos t tan t 7. t 7 corresponds to sin t cos t tan t,.,.. t corresponds to sin cos tan 8. t corresponds to sin t cos t tan t,.,. Houghton Mifflin Compan. All rights reserved. 9. t corresponds to 0,. sin t cos t 0 tan t is undefined.. t corresponds to sin t cos t,. csc t sec t 0. t corresponds to, 0. sin 0 cos tan 0 0 tan t cot t

14 8 Chapter Trigonometric Functions. t corresponds to,.. t corresponds to 0,. sin t cos t sin t cos t 0 csc t sec t is undefined. tan t tan t is undefined. cot t 0 csc t sec t cot t. t corresponds to 0,. sin cos 0 tan 0 csc sec 0 cot 0 0. t 7 corresponds to sin 7 cos 7 tan 7 undefined undefined,. csc 7 sec 7 cot 7. t corresponds to sin t cos t tan t 7. sin sin 0,. csc t sec cot Houghton Mifflin Compan. All rights reserved.

15 Section. Trigonometric Functions: The Unit Circle 8 8. Because 0. Because. Because 7 : cos 7 cos cos 9 : sin 9 sin sin 9 : sin sin 9 sin 9. cos 8 cos. cos. sin 9 cos cos sin. Because 8 : cos 8 cos cos. sin t (a) sint sin t csct csc t. cos t 7. cost (a) cost cos t (a) cos t cost sect cost cos t sect cost Houghton Mifflin Compan. All rights reserved sint 8 (a) sin t sint 8 csc t sint sint 8 cos t (a) cos t cos t 9. sin t (a) sin t sin t sint sin t. sin cost cos t

16 8 Chapter Trigonometric Functions. tan cos sin csc cot cos tan.7 8. cos csc sec.8.0 sin 0.8 cos.8. sec.8.8 cos.8. sin cot..8 tan.. tan.7.0. csc..00 sin.. tan sec..79 cos. 8. csc..9 sin. 9. (a) sin cos (a) sin cos (a) sin t 0. t 0. or (a) sin t 0.7 t.0 or t. 7. cos t 0. cos t 0.7 t.8 or. t 0.7 or t. I e t sin t I0.7 e. sin amperes 7. At t., I e. sin. 0.0 amperes. t et cos t (a) 0 e0 cos0 foot t t cos t (a) 0 cos ft cos ft (c) cos 0.7 ft (c) The maimum displacements are decreasing because of friction, which is modeled b the e t term. (d) et cos t 0 cos t 0 t, t, Houghton Mifflin Compan. All rights reserved.

17 Section. Trigonometric Functions: The Unit Circle False. sin > True 79. False. 0 corresponds to, (a) The points have -ais smmetr. sin t sin t since the have the same -value. (c) cos t cos t since the -values have opposite signs. 80. True 8. (a) The points, and, are smmetric about the origin. Because of the smmetr of the points, ou can make the conjecture that sint sin t. (c) Because of the smmetr of the points, ou can make the conjecture that cost cos t. 8. cos , cos sin0. sin Thus, cos t cos t. sin 0.8 Therefore, sin t sin t sint t. 8. cos cos 8. sin sin sec sec cos cos csc csc tan tan (, ) θ θ cot cot (, ) Houghton Mifflin Compan. All rights reserved. 87. ht f tgt is odd. 88. ft sin t and gt tan t ht f tgt f tgt ht Both f and g are odd functions. ht ftgt sin t tan t ht sint tant sin ttan t sin t tan t ht The function ht ftgt is even.

18 88 Chapter Trigonometric Functions 89. f 9 9 f f 90. f f f 9 9 f f 9. f,, 0, 0 7 f f 0 9. f, >, > 9 f, 0, < f, < 9. f 9. f Asmptotes:, Asmptotes:,, f 0 8, Asmptotes:, Houghton Mifflin Compan. All rights reserved.

19 Section. Right Triangle Trigonometr f 8 Slant asmptote: 7 7 Vertical asmptotes:.08, Domain: all real numbers 0, Intercepts: 0,,, 0,, 0 No asmptotes ln Domain: all 0 ln 0 ± Intercepts:, 0,, 0 Asmptote: f 00. Domain: all real numbers Intercept: 0, Asmptote: f Domain: all real numbers Intercepts: 7 7 7, 0, 0, 7 Asmptotes:, 0 Section. Right Triangle Trigonometr Houghton Mifflin Compan. All rights reserved. You should know the right triangle definition of trigonometric functions. (a) sin opp cos adj hp hp (c) (d) csc hp (e) sec hp opp adj (f) You should know the following identities. (a) sin csc (c) cos csc sin sec (d) sec (e) tan (f) cot cos cot tan (g) (h) cot cos tan sin cos sin (i) sin cos (j) tan sec (k) cot csc and tan opp adj cot adj opp 90, Adjacent side You should know that two acute angles are complementar if and cofunctions of complementar angles are equal. You should know the trigonometric function values of 0,, and 0, or be able to construct triangles from which ou can determine them. θ Hpotenuse Opposite side

20 90 Chapter Trigonometric Functions Vocabular Check. (a) iii vi (c) ii (d) v (e) i (f) iv. hpotenuse, opposite, adjacent. elevation, depression.. θ b θ b 9 adj sin opp hp cos adj hp tan opp adj csc hp opp sec hp adj cot adj opp sin cos tan csc sec opp adj opp hp hp hp hp adj adj opp cot adj opp c θ θ hp 8 7 sin opp hp 8 7 cos adj hp 7 tan opp adj 8 csc hp 7 opp 8 sec hp adj 7 C 8 8 sin cos tan csc sec 8 8 Houghton Mifflin Compan. All rights reserved. cot adj opp 8 cot

21 Section. Right Triangle Trigonometr 9. opp 0 8 sin opp hp 0 0 opp.. sin opp. hp.. θ cos adj hp 8 0 θ 8 cos adj hp. tan opp adj 8 tan opp adj. csc hp opp 0 csc hp. opp. sec hp adj 0 8 sec hp adj. cot adj opp 8 cot adj opp. The function values are the same since the triangles are similar and the corresponding sides are proportional. Houghton Mifflin Compan. All rights reserved.. adj 8 sin cos tan csc sec opp adj hp opp hp hp hp 8 adj 8 opp 8 adj cot adj opp 8 8 θ 8 adj 7. sin cos tan csc sec opp hp 7. 8 adj hp opp hp opp hp 7. adj adj 7. cot adj opp The function values are the same because the triangles are similar, and corresponding sides are proportional. θ

22 9 Chapter Trigonometric Functions 7. adj 8 adj sin opp hp cos adj hp θ sin opp hp cos adj hp θ tan opp adj tan opp adj csc hp opp csc hp opp sec hp adj sec hp adj cot adj opp cot adj opp The function values are the same since the triangles are similar and the corresponding sides are proportional. 8. hp sin cos opp hp adj hp θ hp sin cos θ tan csc sec opp hp opp hp adj adj tan csc sec cot adj opp cot The function values are the same because the triangles are similar, and corresponding sides are proportional. Houghton Mifflin Compan. All rights reserved.

23 Section. Right Triangle Trigonometr 9 9. Given: sin opp hp adj adj cos adj hp tan opp adj cot adj opp sec hp adj θ 0. hp sin cos tan csc opp adj opp hp opp sec hp adj hp hp adj θ csc hp opp. Given: sec opp sin cos opp hp adj θ. opp sin tan csc θ 0 tan sec 7 cot csc cot Houghton Mifflin Compan. All rights reserved.. Given: sin 0 0 cos 0 0 sec 0 cot csc 0 tan opp adj hp 0 hp 0 θ. adj 7 7 sin cos tan sec opp adj hp opp hp 7 adj 7 cos 7 7 cot 7 tan θ 7 7

24 9 Chapter Trigonometric Functions. Given: cot 9 9 hp hp 97 adj opp sin cos θ sin 8 adj 8 cos tan csc adj hp opp 8 adj sin 8 θ 8 tan 9 sec 97 9 csc 97 sec cos cot tan sin 9. tan. cot. cos Function (deg) (rad) Function Value cot 8. cos. sin. tan Function (deg) (rad) Function Value 0. sec. csc 0 7. sin csc 8. cos sec 9. tan cot 0. csc sin. sec cos. cot tan. sin cos. tan sec 9. tan90 cot. sin 0, cos 0 (a) tan 0 sin 0 cos 0 (c) cos 0 sin 0 0. cot90 tan. tan sin cos 7. sin90 cos. sec90 csc sin 0 cos 0 (d) cot 0 cos 0 sin 0. cot cos sin 8. cos90 sin. csc90 sec Houghton Mifflin Compan. All rights reserved.

25 Section. Right Triangle Trigonometr 9. sin 0, tan 0. csc, sec (a) csc 0 sin 0 (a) sin csc (c) cot 0 tan90 0 tan 0 cos 0 (d) cot 0 tan 0 sin 0 tan 0 (c) cos sec tan sin cos (d) sec90º csc. sec, tan (a) cos sec cot tan (c) cot90 tan (d) sin tan cos 7. cos (a) sec cos (c) cot cos sin sin cos sin sin sin (d) sin90 cos Houghton Mifflin Compan. All rights reserved. 8. tan (a) (c) cot tan lies in Quadrant I or III. sec tan cos tan90 cot tan (d) csc cot 9. tan cot tan tan 0. csc tan sin sec sin cos cos. tan cos sin cos cos sin. cot sin cos sin cos sin

26 9 Chapter Trigonometric Functions. cos cos cos. csc cot csc cot csc cot sin cos cos sin. sin cos sin cos cos sin sin cos sin cos csc sec sin cos. tan cot tan tan cot tan tan cot cot cot csc 7. (a) sin 0. cos (a) tan cot tan (a) sec sec. csc 8º 7.99 cos. sin8 0º (a) cos8 0 cos sec8 0 cos cos8 0. Make sure that our calculator is in radian mode. (a) cot.07 tan tan (a) sec. cos cos.. (a). (a) 7. (a) sin csc sec cot csc sin (a). (a) 8. (a) cos tan cos tan cot cos 0 tan sec 0 0 Houghton Mifflin Compan. All rights reserved.

27 Section. Right Triangle Trigonometr tan 0 0 cos 0 0 r 0 tan 0 0 r cos cos 0 cos 0 sin 0 sin 0 cos 0 cos 0 8 sin 0 sin cot cot sin 0 8 r r sin 0 7. tan r 0 0 r 0 tan 0 0 r 0 0 r tan r r 0 Houghton Mifflin Compan. All rights reserved (a) tan r r 8 tan and tan h Thus, h. (c) h h θ. feet 78. (a) 0 7 sin 7 0 (c) 0 sin meters

28 98 Chapter Trigonometric Functions 79. tan opp adj tan 8 w cot 9 c h cot. c h. 9 c h w 00 tan 8 0 feet not drawn to scale Subtracting, h cot. cot 9 h cot. cot miles. 8. (a) (c) tan (a) L L 0 feet 0 0 ftsec ftsec rate down the zip line vertical rate (c) sin feet above sea level Vertical rate sin. 9. feet minutes to reach top feet per minute 8. (, ) (, ) 0 sin 0 sin 0 8 cos 0 cos 0 8, 8, 8 0 sin 0º sin 0 8 cos 0 cos 0 8, 8, 8 Houghton Mifflin Compan. All rights reserved.

29 Section. Right Triangle Trigonometr ,.0 sin tan 0 0. cos cot True sin 0 csc 0 sin 0 sin 0 sec csc False sin cos 87. True cot csc for all 88. No. tan sec, so ou can find ± sec. 89. (a) sin cos Sine and tangent are increasing, cosine is decreasing. (c) In each case, tan sin. cos tan cos cos sin90 and 90 are complementar angles. sin Houghton Mifflin Compan. All rights reserved

30 00 Chapter Trigonometric Functions Section. Trigonometric Functions of An Angle Know the Definitions of Trigonometric Functions of An Angle. If is in standard position,, a point on the terminal side and r 0, then: sin r cos r tan, 0 csc r, 0 sec r, 0 cot, 0 You should know the signs of the trigonometric functions in each quadrant. You should know the trigonometric function values of the quadrant angles 0, You should be able to find reference angles. You should be able to evaluate trigonometric functions of an angle. (Use reference angles.) You should know that the period of sine and cosine is. You should know which trigonometric functions are odd and even. Even: cos and sec Odd: sin, tan, cot, csc,, and. Vocabular Check.. csc.. r. cos. cot 7. reference r. (a),,, 8, r 9 r 7 sin r cos r tan csc r sec r cot sin r cos r tan 8 csc r sec r cot 8 Houghton Mifflin Compan. All rights reserved.

31 Section. Trigonometric Functions of An Angle 0. (a),, r r sin r sin r cos r cos r tan tan csc r csc r sec r sec r cot cot. (a),,,, r r sin r csc r sin r csc r cos r sec r cos r sec r tan cot tan cot. (a),, r 0 r sin r sin r Houghton Mifflin Compan. All rights reserved. cos r tan csc r sec r 0 0 cot 0 cos r tan csc r sec r cot

32 0 Chapter Trigonometric Functions., 7,. 8,, r 9 7 r 8 7 sin r csc r sin r 7 cos r 8 7 cos r 7 sec r 7 tan 8 csc r 7 tan 7 cot 7 r 7 sec cot ,, 8., 0 r 9 r 0 sin r sin r 0 cos r cos r tan tan 0 csc r csc r sec r sec r cot cot 9.,, 0 0., r 00 9 r sin r 9 9 sin r cos r tan csc r sec r cot cos tan csc sec r r r cot Houghton Mifflin Compan. All rights reserved.

33 Section. Trigonometric Functions of An Angle 0., 0, 8. r 0 8 sin r 8 cos r tan 8 0 csc r sec r 0, 9 r sin cos tan csc sec r r r r cot cot. sin < 0 lies in Quadrant III or Quadrant IV. cos < 0 sin < 0 and lies in Quadrant II or Quadrant III. cos < 0 lies in Quadrant III.. sec > 0 and r > 0 and Quadrant IV < 0 cot < 0. cot > 0 lies in Quadrant I or Quadrant III. cos > 0 lies in Quadrant I or Quadrant IV. cot > 0 and cos > 0 lies in Quadrant I.. tan > 0 and > 0 and Quadrant III r < 0 csc < 0 7. sin r 9 8. cos r in Quadrant II in Quadrant III Houghton Mifflin Compan. All rights reserved. sin r cos r tan csc r sec r cot sin cos r r csc sec tan cot

34 0 Chapter Trigonometric Functions 9. sin < 0 < 0 0. csc tan sin r 8 7 cos r 8 7 cot 8 r 7 csc r 7 sec r 7 8 cot sin cos r < 0 r r ± csc sec tan cot. sec r sin 0 sin r cos r tan csc r sec r cot. sin 0 and csc is undefined. sec tan sin 0 cos cot is undefined. cos cos. cot is undefined sin r 0 n., 0, r csc r is undefined.. tan is undefined and sin cos 0 tan is undefined.. cos r r r sec r csc sec is undefined. tan 0 0 cot is undefined. cot 0. To find a point on the terminal side of, use an point on the line that lies in Quadrant II., is one such point.,, r sin cos tan csc sec cot Houghton Mifflin Compan. All rights reserved.

35 Section. Trigonometric Functions of An Angle 0,. Quadrant III, > 0 r 9 0 sin cos tan csc sec r r r r cot To find a point on the terminal side of, use an point on the line that lies in Quadrant III., is one such point.,, r sin cos tan csc sec cot 8. 0, Quadrant IV, > 0 r 9 Houghton Mifflin Compan. All rights reserved. sin cos r r csc sec tan cot 9.,, 0 0. tan undefined., 0, 0 sec r 0 cot since corresponds to 0,. 0.,, 0 cot 0 undefined. csc 0 r undefined.,, 0. csc 0 sec 0 r sin. csc sin

36 0 Chapter Trigonometric Functions θ = 0 θ =0 θ = θ = 9. is coterminal with θ = θ = 0 θ = 0 θ =.,. θ = π π θ = θ = π θ = π 7. is coterminal with π θ = θ = π 08. θ = 08 θ = 8 θ = π θ = π 0 8 θ = θ = 8 Houghton Mifflin Compan. All rights reserved.

37 Section. Trigonometric Functions of An Angle lies in θ = 9 θ = 8 Quadrant III. Reference angle: 80 θ = θ = 9. is coterminal with θ = π 7 θ = π θ = 7π 7 θ = π.8. lies in Quadrant III. Reference angle:.8.. lies in Quadrant IV. Reference angle:..8 θ =. θ =.8 θ =.8 θ =. Houghton Mifflin Compan. All rights reserved.,. Quadrant III sin sin cos cos tan tan 70. is coterminal with 0, Quadrant IV sin70 sin 0 cos70 cos 0 tan70 tan 0 00, ,. Quadrant IV sin 00 sin 0 cos 00 cos 0 tan 00 tan 0 9,,. Quadrant III sin9 sin cos9 cos tan9 tan

38 08 Chapter Trigonometric Functions 7. Quadrant IV, 8. sin cos tan sin cos tan sin cos, tan 9. Quadrant IV, sin cos tan sin cos tan. Quadrant II, sin sin cos cos tan tan 7 7 sin 7 cos 7 tan 7 Quadrant III, sin cos tan 7. is coterminal with. 0., sin cos tan 0 sin cos 0 in Quadrant III. 0 sin tan 0 tan cos. is coterminal with.. 0 sin 0 cos 0, 0 tan Houghton Mifflin Compan. All rights reserved.

39 Section. Trigonometric Functions of An Angle 09. sin. cot sin cos cos sin cos cot csc csc 0 csc csc > 0 in Quadrant II. 0 csc cos > 0 cos 9 cos in Quadrant IV. csc sin sin 0 csc 0 0 cos 7. csc 8. cos 8 cot csc cot csc cot cot cos sec sec cos sec 8 8 cot < 0 in Quadrant IV. cot 9. sec tan cot Houghton Mifflin Compan. All rights reserved. tan sec tan sec tan 9 tan tan > 0 in Quadrant III. tan cot csc csc csc ± Quadrant IV csc

40 0 Chapter Trigonometric Functions 7. sin is in Quadrant II. and cos < 0 cos sin tan sin cs csc sin sec cos cot tan 7. cos is in Quadrant III. 7 and sin < 0 sin cos 9 9 tan sin 07 0 cs 7 cot 0 tan 0 0 csc 7 70 sin 0 0 sec cos tan and cos < 0 is in Quadrant II. sec tan 7 cos 7 sec 7 7 sin tan cos csc 7 7 sin 7 cot tan 7. cot and sin > 0 is in Quadrant II. tan cot sec tan cos sec sin tan cos csc sin Houghton Mifflin Compan. All rights reserved.

41 Section. Trigonometric Functions of An Angle 7. csc is in Quadrant IV. and tan < 0 sin csc cos sin 9 sec cos tan sin cos cot tan 7. sec is in Quadrant III. and cot > 0 cos sec sin cos 9 7 csc 7 sin 7 7 tan sin 7 cos 7 cot tan sin sec tan. cos 80. csc 0 8. cos0 0.0 sin cot0 tan sec80 cos80 8. csc sin tan tan csc cos sin Houghton Mifflin Compan. All rights reserved. 89. (a) sin reference angle is 0 or 90. (a) cos reference angle is º or and is in Quadrant I or Quadrant II. Values in degrees: Values in radian: sin reference angle is 0 or and is in Quadrant III or Quadrant IV. Values in degrees: 0, 0, 0, 0 Values in radians: 7, and is in Quadrant I or IV. Values in degrees: Values in radians: cos reference angle is or and Values in degrees:,, 7 is in Quadrant II or III., Values in radians:,

42 Chapter Trigonometric Functions 9. (a) csc reference angle is 0 or and is in Quadrant I or Quadrant II. Values in degrees: Values in radians: cot reference angle is or and is in Quadrant II or Quadrant IV. Values in degrees: 0, 0,, Values in radians:, 7 9. (a) csc Reference angle is or. csc Values in degrees: Values in radians: Reference angle is or 0. Values in degrees: Values in radians:,, 7 sin 0, 0 sin, 9. (a) sec reference angle is or 0, and is in Quadrant II or Quadrant III. Values in degrees: 0, 0 Values in radians: cos reference angle is or 0, and is in Quadrant II or Quadrant III., 7 Values in degrees: 0, 0 Values in radians:, 9. (a) cot cos sin Reference angle is or 0. Values in degrees: 0, 0 Values in radians: Values in degrees: or Values in radians:, or 7 9. (a) 9. (a) f g sin 0 cos 0 cos 0 sin 0 (c) cos 0 cos 0 sin 0 (c) cos 0 f g sin 0 cos 0 (d) (e) sin 0 cos 0 sin 0 (f) cos0 cos 0 (d) (e) sin 0 cos 0 sin 0 (f) cos0 cos 0 Houghton Mifflin Compan. All rights reserved.

43 Section. Trigonometric Functions of An Angle 97. (a) f g sin cos cos sin (c) cos 0 (d) (e) sin cos sin (f) cos cos 98. (a) (c) 99. (a) f g sin cos cos sin cos 0 (d) sin cos (c) f g sin 0 cos 0 cos 0 sin 0 cos 0 (d) sin 0 cos 0 (e) sin (f) cos cos (e) sin 0 (f ) cos0 cos0 Houghton Mifflin Compan. All rights reserved. 00. (a) (c) 0. (a) f g sin 00 cos 00 cos 00 sin 00 cos 00 (d) sin 00 cos 00 (c) f g sin 7 7 cos cos 7 7 sin 7 cos (d) sin 7 7 cos (e) sin 00 (f) cos00 cos00 (e) sin 7 (f) cos 7 cos 7

44 Chapter Trigonometric Functions 0. (a) (c) f g sin cos cos sin cos (d) sin cos (e) sin (f) cos cos 0. (a) (c) f g sin cos sin cos (d) sin cos cos (e) sin (f) cos cos 0. (a) (c) f g sin cos sin cos (d) sin cos cos (e) sin (f) cos cos 0. (a) 0. (a) f g sin 70 cos 70 0 (d) cos 70 sin 70 0 (c) cos (a) (c) cos 80 f g sin 80 cos 80 0 (d) cos 80 sin 80 0 f g sin 7 cos 7 7 sin 0 (c) cos (e) sin 70 cos sin 70 (f) cos70 cos70 0 (e) 7 cos 0 (d) sin 80 cos sin (f) cos80 cos80 (e) sin 7 7 cos 0 0 sin 7 (f) cos 7 cos 7 0 Houghton Mifflin Compan. All rights reserved.

45 Section. Trigonometric Functions of An Angle 08. (a) f g sin cos sin 0 (c) cos 0 0 cos 0 (d) (e) sin cos 0 0 sin (f) cos cos T cos t 7 (a) Januar: Jul: t T cos 7 9 t 7 T 70 (c) December: t T.7 S. 0.t. sin t, t Jan. 00 (a) (c) (d) S. 0.. sin S.0 thousand S 7. thousand S.8 thousand Answers will var..7 thousand Houghton Mifflin Compan. All rights reserved.. sin (a) (c) d d 0 d sin 0 90 d sin 90 0 d sin.9 sin 0 miles miles miles. True. The reference angle for is and sine is positive in Quadrants I and II. 80 9,. As increases from 0 to 90, decreases from cm to 0 cm and increases from 0 cm to cm. Therefore, sin increases from 0 to, and cos decreases from to 0. Thus, tan begins at 0 and increases without bound. When the tangent is undefined. 90,. False. cot and cot

46 Chapter Trigonometric Functions. (a) sin sin It appears that sin sin80.. Function Domain Range Evenness No Yes No Oddness Yes No Yes Period sin cos tan,,,, n Zeros n n All reals ecept n, Function csc sec cot Domain All reals ecept All reals ecept n All reals ecept Range,,,,, Evenness No Yes No Oddness Yes No Yes Period n Zeros None None Patterns and conclusions ma var. n n ± 0 ±.9, , ± ± Houghton Mifflin Compan. All rights reserved.

47 Section. Graphs of Sine and Cosine Functions ± 9.908,.908 ±7 0 ( 0 etraneous). 7 log 7 log 7 ln 7 ln e 90 e 8 e ln 8 ln 8.7. ln e ln 0 ln 0 ln 0 0 e e 0. Section. Graphs of Sine and Cosine Functions Houghton Mifflin Compan. All rights reserved. You should be able to graph a sinb c and a cosb c. Amplitude: b a Shift: Solve b c 0 and b c. Ke increments: (period) Vocabular Check. amplitude. one ccle.. phase shift b

48 8 Chapter Trigonometric Functions.. f sin (a) -intercepts:, 0,, 0, 0, 0,, 0,, 0 -intercept: 0, 0 (c) Increasing on:, Decreasing on:, (d) Relative maima: (a) Relative minima: f cos -intercepts: -intercept: 0,,,,,,,,,,, (c) Increasing on:, 0,, Decreasing on:,, 0, (d) Relative maima:,, 0,,, Relative minima:,,,,,, 0,, 0,, 0,, 0. sin Amplitude: Xmin = - Xma = Xscl = Ymin = - Yma = Yscl =. cos b Amplitude: a Xmin = - Xma = Xscl = Ymin = - Yma = Yscl =. 7. cos Amplitude: sin Xmin = - Xma = Xscl = Ymin = - Yma = Yscl =. 8. sin b Amplitude: a cos b Xmin = - Xma = Xscl = Ymin = - Yma = Yscl = Houghton Mifflin Compan. All rights reserved. Amplitude: Amplitude: a

49 Section. Graphs of Sine and Cosine Functions 9 9. sin 0. Amplitude: cos b Amplitude: a. cos Amplitude:. cos. sin. cos b 8 b Amplitude: a Amplitude: Amplitude: a. f sin g sin The graph of g is a horizontal shift to the right units of the graph of f (a phase shift).. f cos, g cos g is a horizontal shift of f units to the left. 7. f cos g cos The graph of g is a reflection in the -ais of the graph of f. 8. f sin, g sin g is a reflection of f about the -ais. (or, about the -ais) 9. f cos g cos The graph of g has five times the amplitude of f, and reflected in the -ais. 0. f sin, g sin The amplitude of g is one-half that of f. g is a reflection of f in the -ais. Houghton Mifflin Compan. All rights reserved.. f sin g sin The graph of g is a vertical shift upward of five units of the graph of f.. The graph of g has twice the amplitude as the graph of f. The period is the same.. The graph of g is a horizontal shift units to the right of the graph of f.. f cos, g cos g is a vertical shift of f si units downward.. The period of g is one-half the period of f.. Shift the graph of f two units upward to obtain the graph of g.

50 0 Chapter Trigonometric Functions 7. f sin 8. f sin Amplitude: g sin g sin Amplitude: 0 sin g sin 0 π f π g f π 9. f cos 0. f cos Amplitude: g cos is a vertical shift of the graph of f four units upward. g cos 0 cos 0 0 g cos. π f sin Amplitude: f π g is the graph of f sin shifted verticall three units upward. π π g f π π π f g π Houghton Mifflin Compan. All rights reserved.

51 Section. Graphs of Sine and Cosine Functions. f sin. f cos g sin 0 f Amplitude: g cos is the graph of f shifted units to the left. g f f g g π π. f cos g cos 0. f sin, g cos sin cos Amplitude: cos cos f = g f g π π Houghton Mifflin Compan. All rights reserved.. f sin, g cos 0 sin π cos π f = g Conjecture: sin cos

52 Chapter Trigonometric Functions 7. f cos 8. f cos, g cos g sin Thus, f g. f = g sin cos 0 cos cos f = g Conjecture: cos cos 9. sin 0. cos Amplitude: Ke points: 0, 0,, 0,,,,,, 0 Amplitude: π π π π 0. π 0. π π π. cos. sin Amplitude: Ke points: 0,,, 0,,,, 0,, π π Amplitude: π π π π Houghton Mifflin Compan. All rights reserved.

53 Section. Graphs of Sine and Cosine Functions. sin Amplitude: Shift: Set and 0 Ke points: ; a, b, c 9, 0,,,, 0, 7,, 9, 0 π π. sin Horizontal shift units to the right π π π. 8 cos 0 8 Amplitude: 8 Ke points:, 8,, 0, 0, 8,, 0,, 8 π π 8 0 π π. cos 7. sin Amplitude: Amplitude: Houghton Mifflin Compan. All rights reserved. Horizontal shift units to the left Note: cos π π π sin

54 Chapter Trigonometric Functions 8. 0 cos 9. cos t 0. sin Amplitude: 0 Amplitude: Amplitude: cos. Amplitude: cos. Amplitude: sin Amplitude: π π π π. sin Amplitude: 8. cos. Amplitude: cos Amplitude: 7. 8 sin cos Amplitude: Amplitude: sin 0t Amplitude: Houghton Mifflin Compan. All rights reserved.

55 Section. Graphs of Sine and Cosine Functions cos0 t. Amplitude: f a cos d Amplitude: 8 0 Since f is the graph of g cos reflected about the -ais and shifted verticall four units upward, we have a and d. Thus, f cos cos.. f a cos d Amplitude: Reflected in the -ais: a d cos 0 d cos. f a cos d. a cos d. Amplitude: 7 Graph of f is the graph of g cos reflected about the -ais and shifted verticall one unit upward. Thus, f cos. Amplitude: Reflected in -ais, a d cos f a sinb c Amplitude: Since the graph is reflected about the -ais, we have a. a b b Phase shift: c 0 Thus, f sin.. a sinb c 7. f a sinb c Houghton Mifflin Compan. All rights reserved. 8. Amplitude: a b b Phase shift: sin c 0 a sinb c 9. Amplitude: a b Phase shift: c b sin c b Amplitude: Phase shift: a b Thus, f sin sin c 0 c. b c 0 when. In the interval,, sin when,, 7,.

56 Chapter Trigonometric Functions 70. cos when,. 7. v 0.8 sin (a) t 0 Time for one ccle one period sec (c) Ccles per min 0 0 ccles per min (d) The period would change. 7. S cos (a) 0 t 7. h sin t 7 0 (a) Maimum sales: December t Minimum sales: June t Minimum: 0 feet Maimum: 0 feet P 00 0 cos 8 t (a). 8 heartbeat C 0.. sin t 0.9 das b This is to be epected: das ear The constant 0. gallons is the average dail fuel consumption. heartbeatssecond 80 heartbeatsminute (c) Consumption eceeds 0 gallonsda when. (Graph C together with 0.) (Beginning of Ma through part of September) Houghton Mifflin Compan. All rights reserved.

57 Section. Graphs of Sine and Cosine Functions 7 7. (a) Yes, is a function of t because for each value of t there corresponds one and onl one value of. The period is approimatel seconds. (c) One model is (d) 0. sin t. The amplitude is approimatel.. 0. centimeters (a) (c) sin The model is a good fit. (d) The period is (e) June 9, 007 is da. Using the model, or 7.09%. Houghton Mifflin Compan. All rights reserved. 78. (a) (c) At 9.7 sin0.7t The model somewhat fits the data. 9 0 The model is a good fit. 79. True. The period is True (d) Nantucket: Athens: The constant term (d) gives the average dail high temperature. (e) Period for Period for Nt At 0.7 You would epect the period to be ( ear). (f) Athens has greater variabilit. This is given b the amplitude. 80. False. The amplitude is that of cos. 8. Answers will var. 8. The graph passes through 0, 0 and has period. 8. The amplitude is and the period. Since Matches (e). 0, is on the graph, matches (a). 8. The period is and the amplitude is. Since 0, and, 0 are on the graph, matches (c). 8. The period is. Since 0, is on the graph, matches (d).

58 8 Chapter Trigonometric Functions 87. (a) h cos is even. h sin is even. (c) h sin cos is odd. h() = cos h() = sin h() = sin cos 88. (a) In Eercise 87, f cos is even and we saw that h cos is even. Therefore, for f even and h f, we make the conjecture that h is even. In Eercise 87, g sin is odd and we saw that h sin is even. Therefore, for g odd and h g, we make the conjecture that h is even. (c) From part (c) of 87, we conjecture that the product of an even function and an odd function is odd. 89. (a) sin sin Undef As 0, f sin approaches. sin (c) As approaches 0, approaches. 90. (a) cos (a) π π cos Undef π π (c) Net term for sine approimation: Net term for cosine approimation: π π 0. cos As 0, approaches 0. cos (c) As approaches 0, approaches 0. π 7 7!! π Houghton Mifflin Compan. All rights reserved.

59 Section. Graphs of Other Trigonometric Functions 9 9. (a) sin 0.8 (d) cos sin (e) (c) sin 0.87 (f ) cos cos In all cases, the approimations are ver accurate Slope 7 0 (0, ) (, 7) 9. (, ) (, ) m Answers will var. (Make a Decision) Section. Graphs of Other Trigonometric Functions Houghton Mifflin Compan. All rights reserved. You should be able to graph: a tanb c a cotb c a secb c a cscb c When graphing a secb c or a cscb c ou should know to first graph a cosb c or a sinb c since (a) The intercepts of sine and cosine are vertical asmptotes of cosecant and secant. The maimums of sine and cosine are local minimums of cosecant and secant. (c) The minimums of sine and cosine are local maimums of cosecant and secant. You should be able to graph using a damping factor. Vocabular Check. vertical. reciprocal. damping