Chapter 7 Analytic Trigonometry

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1 Chapter 7 Analytic Trigonometry Section 7.. Domain: { is any real number} ; Range: { y y }. { } or { }. [, ). True. ;. ; 7. sin y False. The domain of. True. True.. y sin is. sin 0 We are finding the angle,, whose sine equals 0. 0, 0 sin 0 0 cos We are finding the angle, 0, whose cosine equals., 0 0 cos 0. sin ( ) We are finding the angle,, whose sine equals., sin ( ). cos ( ) 7. We are finding the angle, 0, whose cosine equals., 0 cos tan 0 We are finding the angle, < <, whose tangent equals 0. tan 0, < < 0 tan tan ( ) We are finding the angle, < <, whose tangent equals. tan, < < tan ( ) 8 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

2 Section 7.: The Inverse Sine, Cosine, and Tangent Functions 9. sin We are finding the angle,, whose sine equals. sin, sin. sin We are finding the angle,, whose sine equals. sin, sin 0. tan We are finding the angle, < <, whose tangent equals. tan, < < tan. cos We are finding the angle, 0, whose cosine equals cos. cos, 0. tan We are finding the angle, < <, whose tangent equals. tan, < < tan. sin We are finding the angle,, whose sine equals sin. sin, 9 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

3 Chapter 7: Analytic Trigonometry sin cos tan.7 tan cos sin 0. 8 tan ( 0.) 0.8 tan. sin ( 0.) 0. cos ( 0.).0 cos.08 sin 0. cos cos follows the form of the equation ( ) cos f f ( cos ). Since is in the interval 0,, we can apply the equation directly and get cos cos. sin sin 0 follows the form of the f f sin sin. Since equation ( ) is in the interval, 0 the equation directly and get sin sin 0. 0, we can apply tan tan 8 follows the form of the equation f ( f ) tan ( tan ). Since is in the interval, 8, we can apply the equation directly and get tan tan 8. 8 sin sin 7 follows the form of the equation f ( f ) sin ( sin ). Since is in the interval, 7, we can apply the equation directly and get sin sin sin sin 8 follows the form of the equation f ( f ) sin ( sin ), but we cannot use the formula directly since 9 is not 8 in the interval,. We need to find an angle in the interval, for which 9 sin. The angle 9 is in quadrant III 8 8 so sine is negative. The reference angle of 9 is 8 and we want to be in quadrant IV so sine 8 will still be negative. Thus, we have 9 sin sin 8 8. Since is in the interval 8,, we can apply the equation above and 9 get sin sin sin sin Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

4 Section 7.: The Inverse Sine, Cosine, and Tangent Functions.. cos cos follows the form of the equation ( ) cos f f ( cos ), but we cannot use the formula directly since is not in the interval 0,. We need to find an angle in the interval 0, for which cos. The angle is in quadrant I so the reference angle of is. Thus, we have cos cos. Since is in the interval 0,, we can apply the equation above and get cos cos cos cos. tan tan follows the form of the equation f ( f ) tan ( tan ), but we cannot use the formula directly since is not in the interval,. We need to find an angle in the interval, for which tan tan. The angle is in quadrant II so tangent is negative. The reference angle of is and we want to be in quadrant IV so tangent will still be negative. Thus, we have tan tan. Since is in the interval,, we can apply the equation above and get tan tan tan tan.... tan tan follows the form of the equation tan ( tan ) f f. but we cannot use the formula directly since is not in the interval,. We need to find an angle in the interval, for which tan tan. The angle is in quadrant III so tangent is positive. The reference angle of is and we want to be in quadrant I so tangent will still be positive. Thus, we have tan tan. Since is in the interval,, we can apply the equation above and get tan tan tan tan. sin sin follows the form of the equation ( ) sin sin f f. Since is in the interval,, we can apply the equation directly and get sin sin. cos cos follows the form of the equation cos( cos ) f f. Since is in the interval,, we can apply the equation directly and get cos cos. Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

5 Chapter 7: Analytic Trigonometry 7. tan ( tan ) ( ) tan ( tan ) follows the form of the equation f f. Since is a real number, we can apply the equation directly tan tan. and get ( ) tan ( tan ) 8. tan tan follows the form of the equation f f. Since is a real number, we can apply the equation directly tan tan. and get 9. Since there is no angle such that., the quantity cos. is not defined. Thus, cos cos. is not defined. 0. Since there is no angle such that, sin is not defined. Thus, the quantity sin sin is not defined.. tan ( tan ) ( ) tan ( tan ) follows the form of the equation f f. Since is a real number, we can apply the equation directly tan tan. and get. Since there is no angle such that., sin. is not defined. Thus, the quantity sin sin (.). f sin + y sin + is not defined. sin y+ sin y sin y y sin f The domain of f ( ) equals the range of f and is or, in f interval notation. To find the domain of we note that the argument of the inverse sine function is and that it must lie in the interval,. That is, 7 f 7, or The domain of is { }, 7 in interval notation. Recall that the domain of a function equals the range of its inverse and the range of a function equals the domain of its inverse. Thus, the range of f is also, 7.. f tan y tan tan y tany + + tan y + y tan f The domain of f equals the range of f and is < < or, in interval notation. To find the domain of f we note that the argument of the inverse tangent function can be any real number. Thus, the domain of f is all real numbers, or (, ) in interval notation. Recall that the domain of a function equals the range of its inverse and the range of a function equals the domain of its inverse. Thus, the range of f is (, ).. f cos y cos Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

6 Section 7.: The Inverse Sine, Cosine, and Tangent Functions ( y) cos cos ( y) y cos cos y f The domain of f ( ) equals the range of f and is 0, or 0, in interval notation. To find the domain of f we note that the argument of the inverse cosine function is and that it must lie in the interval,. That is, The domain of f is { }, or, in interval notation. Recall that the domain of a function equals the range of its inverse and the range of a function equals the domain of its inverse. Thus, the range of f is,.. f sin y sin sin( y) sin ( y) y sin sin y f The domain of f ( ) equals the range of f and is, or, in f we note that the argument of the inverse sine interval notation. To find the domain of function is and that it must lie in the interval,. That is, f The domain of is { }, or, in interval notation. Recall that the domain of a function equals the range of its inverse and the range of a function equals the domain of its inverse. Thus, the range of f is,. 7. f tan ( + ) y tan ( + ) tan ( y+ ) tan ( y+ ) y+ tan y + tan tan ( + ) f (note here we used the fact that odd function). y tan is an The domain of f ( ) equals the range of f and is, or, in interval notation. To find the domain of f we note that the argument of the inverse tangent function can be any real f is all real number. Thus, the domain of numbers, or (, ) in interval notation. Recall that the domain of a function equals the range of its inverse and the range of a function equals the domain of its inverse. Thus, the range of f is (, ). 8. f cos( + ) + y cos( + ) + ( y+ ) + cos( y+ ) y+ cos y ( ) cos cos Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

7 Chapter 7: Analytic Trigonometry The domain of f ( ) equals the range of f and is, or, in interval notation. To find the domain of f we note that the argument of the inverse cosine function is and that it must lie in the interval,. That is, 0 The domain of f is { 0 }, or 0, in interval notation. Recall that the domain of a function equals the range of its inverse and the range of a function equals the domain of its inverse. Thus, the range of f is 0,. 9. f sin( + ) y sin( + ) sin( y+ ) sin ( y + ) y + sin y sin y sin f The domain of f ( ) equals the range of f and is +, or, + in interval notation. To find f we note that the argument the domain of of the inverse sine function is and that it must domain of its inverse. Thus, the range of f is,. 0. f cos( + ) y cos( + ) cos( y+ ) cos( y + ) y + cos y cos y cos f The domain of f ( ) equals the range of f and is +, or, + in interval notation. To find the f we note that the argument of domain of the inverse cosine function is and that it must lie in the interval,. That is, The domain of f is { }, or, in interval notation. Recall that the domain of a function equals the range of its inverse and the range of a function equals the domain of its inverse. Thus, the range of f is,. lie in the interval,. That is, f, or The domain of is { }, in interval notation. Recall that the domain of a function equals the range of its inverse and the range of a function equals the. sin sin sin The solution set is. Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

8 Section 7.: The Inverse Sine, Cosine, and Tangent Functions. cos cos cos 0 The solution set is {0}. cos cos cos The solution set is... sin.. sin sin The solution set is. tan tan tan The solution set is { }. tan tan tan The solution set is { } cos cos cos 0 cos cos cos The solution set is { }. sin sin sin sin sin The solution set is. 9. Note that cos tan. tan 9.7 a. D.9 hours or hours, minutes b. c. ( ( ) ) ( ( ) ( ) ) cos tan 0 tan 9.7 D hours ( ( ) ( ) ) cos tan.8 tan 9.7 D.8 hours or hours, minutes 70. Note that cos tan. tan 0.7 a. D.9 hours or hours, minutes b. c. ( ( ) ) ( ( ) ( ) ) cos tan 0 tan 0.7 D hours ( ( ) ( ) ) cos tan.8 tan 0.7 D.8 hours or hours, 0 minutes Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

9 Chapter 7: Analytic Trigonometry 7. Note that 8.. cos tan. tan. a. D.0 hours or hours, 8 minutes b. c. ( ( ) ) ( ( ) ( ) ) cos tan 0 tan. D hours ( ( ) ( ) ) cos tan.8 tan. D. hours or hours, minutes 7. Note that 0.7. cos tan. tan.7 a. D 8.9 hours or 8 hours, 7 minutes b. c. 7. a. b. c. ( ( ) ) ( ( ) ( ) ) cos tan 0 tan.7 D hours ( ( ) ( ) ) cos tan.8 tan.7 D 8. hours or 8 hours, 8 minutes ( ( ) ( ) ) cos tan. tan 0 D hours ( ( ) ( ) ) cos tan 0 tan 0 D hours ( ( ) ( ) ) cos tan.8 tan 0 D hours d. There are approimately hours of daylight every day at the equator. 7. Note that 0.. cos tan. tan. a. D hours ( ( ) ) b. c. ( ( ) ( ) ) cos tan 0 tan. D hours ( ( ) ( ) ) cos tan.8 tan. D.0 hours or hours, minute d. The amount of daylight at this location on the winter solstice is 0 hours. That is, on the winter solstice, there is no daylight. In general, for a location at 0' north latitude, it ranges from around-the-clock daylight to no daylight at all. 7. Let point C represent the point on the Earth s ais at the same latitude as Cadillac Mountain, and arrange the figure so that segment CQ lies along the -ais (see figure). y C P s 70 mi D (, y ) Q (70,0) At the latitude of Cadillac Mountain, the effective radius of the earth is 70 miles. If point D(, y) represents the peak of Cadillac Mountain, then the length of segment PD is mile 0 ft 0.9 mile. Therefore, the 80 feet point Dy (, ) (70, y) lies on a circle with radius r 70.9 miles. We now have 70 r cos 0.0 radians 70.9 Finally, s r 70(0.0) 9. miles, (70) 9. and, so t (9.) t hours. minutes (70) Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

10 Section 7.: The Inverse Sine, Cosine, and Tangent Functions Therefore, a person atop Cadillac Mountain will see the first rays of sunlight about. minutes sooner than a person standing below at sea level. 7. a. tan tan. 0 tan tan. 0 0 If you sit 0 feet from the screen, then the viewing angle is about.. tan tan. If you sit feet from the screen, then the viewing angle is about.. ( 0) tan tan If you sit 0 feet from the screen, then the viewing angle is about.8. b. Let r the row that result in the largest viewing angle. Looking ahead to part (c), we see that the maimum viewing angle occurs when the distance from the screen is about. feet. Thus, + ( r ). + r. r. r. Sitting in the th row should provide the largest viewing angle. c. Set the graphing calculator in degree mode and let Y tan tan : The maimum viewing angle will occur when. feet. 77. a. a 0 ; b ; The area is: tan b tan a tan tan 0 0 square units b. 78. a. a 0 ; a ; b ; The area is: tan b tan a tan tan square units b ; The area is: 0 square units sin b sin a sin sin Use MAXIMUM: 7 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

11 Chapter 7: Analytic Trigonometry b. a ; b ; The area is: sin b sin a sin sin square units 79. Here we have α 0', 87 7 ', α 8', and 7 0'. Converting minutes to degrees gives 7 α ( ), ( 87 0 ) ( 7 ), α., and. Substituting these values, and r 90, into our equation gives d 0 miles. The distance from Chicago to Honolulu is about 0 miles. (remember that S and W angles are negative) 80. Here we have α 8', 7 0', α 7 7 ', and 8'. Converting minutes to degrees gives α., 7 ( 7 ), ( 7 0 ) 9 ( 0 ) α, and. Substituting these values, and r 90, into our equation gives d 8 miles. The distance from Honolulu to Melbourne is about 8 miles. (remember that S and W angles are negative) Section 7.. Domain: odd integer multiples of, y y or y Range: { }. True.. sec y,, 0, 8. True cos sin Find the angle,, whose sine equals. sin, cos sin cos sin cos Find the angle, 0, whose cosine equals. cos, 0 sin cos sin tan cos Find the angle, 0, whose cosine equals. tan cos cos, 0 tan. cosine. False 7. True 8 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

12 Section 7.: The Inverse Trigonometric Functions [Continued]... tan sin Find the angle,, whose sine equals. sin, tan sin tan seccos Find the angle, 0, whose cosine equals. cos, 0 sec cos sec cot sin Find the angle,, whose sine equals. sin, cot sin cot 9. csc( tan ) Find the angle, < <, whose tangent equals. tan, < < csc( tan ) csc. sec( tan ) Find the angle, < <, whose tangent equals. tan, < < sec( tan ) sec Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall sin tan Find the angle, < <, whose tangent equals. tan, < < sin tan ( ) sin cos sin Find the angle,, whose sine equals. sin, cos sin cos

13 Chapter 7: Analytic Trigonometry sec sin Find the angle,, whose sine equals. sin, sec sin sec csc cos Find the angle, 0, whose cosine equals. 0 csc cos csc cos sin cos Find the angle, 0, whose cosine equals. cos, 0 cos sin.. tan cot tan Find the angle, < <, whose tangent equals. tan, < < tan cot 7 sin cos sin Find the angle,, whose sine equals sin. sin, 7 cos Find the angle, 0, whose cosine. cos tan cos equals. cos, 0 tan 0 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

14 Section 7.: The Inverse Trigonometric Functions [Continued].. 7. tan sin Let sin. Since and, is in quadrant I, and we let y and r. Solve for : ± 8 ± Since is in quadrant I,. y tan sin tan tan cos Let cos. Since and 0, is in quadrant I, and we let and r. Solve for y: + y 9 y 8 y ± 8 ± Since is in quadrant I, y. y tan cos tan sectan Let tan. Since tan and < <, is in quadrant I, and we let and y. Solve for r: + r r r is in quadrant I. r sectan sec cos sin Let sin. Since and, is in quadrant I, and we let y and r. Solve for : ± 7 Since is in quadrant I, 7. 7 cos sin r cot sin Let sin. Since and, is in quadrant IV, and we let y and r. Solve for : ± 7 Since is in quadrant IV, 7. 7 cot sin cot y csc tan ( ) Let tan ( ). Since tan and < <, is in quadrant IV, and we let and y. Solve for r: + r r r ± Since is in quadrant IV, r. r csc tan ( ) csc y Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

15 Chapter 7: Analytic Trigonometry... sin tan Let tan ( ). Since tan and < <, is in quadrant IV, and we let and y. Solve for r: + 9 r r 0 r ± 0 Since is in quadrant IV, r 0. y sin tan ( ) r cot cos Let cos. Since and 0, is in quadrant II, and we let and r. Solve for y: + y 9 y y ± Since is in quadrant II, y. cot cos cot y sec sin Let sin. Since and, is in quadrant I, and we let y and r. Solve for : ± Since is in quadrant I,. r sec sin sec csctan Let tan. Since tan and < <, is in quadrant I, and we let and y. Solve for r: + r r r is in quadrant I. r csctan csc y sin cos sin 7 cos sin cos cot We are finding the angle, 0 < <, whose cotangent equals. cot, 0 < < cot cot We are finding the angle, 0 < <, whose cotangent equals. cot, 0 < < cot Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

16 Section 7.: The Inverse Trigonometric Functions [Continued] csc ( ) We are finding the angle,, 0, whose cosecant equals. csc,, 0 csc ( ) csc We are finding the angle,, 0, whose cosecant equals. csc,, 0 csc sec We are finding the angle, 0,, whose secant equals. sec, 0, sec. sec ( ) We are finding the angle, 0 whose secant equals. sec sec, 0, ( ),,... cot We are finding the angle, 0 < <, whose cotangent equals cot. cot, 0 < < csc We are finding the angle, 0, whose cosecant equals csc,. csc,, 0 sec cos We seek the angle, 0, whose cosine equals. Now, so lies in quadrant I. The calculator yields cos., which is sec.. an angle in quadrant I, so Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

17 Chapter 7: Analytic Trigonometry. csc sin We seek the angle,, whose sine equals. Now, so lies in quadrant I. The calculator yields sin 0.0, which is an angle in quadrant I, so csc 0.0. csc sin We seek the angle,, whose sine equals. Now, so lies in quadrant IV. The calculator yields sin 0., which is an angle in 9. csc 0.. quadrant IV, so 7. cot tan We seek the angle, 0, whose tangent equals. Now tan, so lies in quadrant I. The calculator yields an 0., which is an angle in quadrant I, so cot cot tan We seek the angle, 0, whose tangent equals. Now tan, so lies in quadrant II. The calculator yields tan., which is an angle in quadrant IV. Since lies in quadrant II,.+.0. Therefore, cot sec ( ) cos We seek the angle, 0, whose cosine equals. Now, lies in quadrant II. The calculator yields cos.9, which is an angle in sec.9. quadrant II, so cot tan We seek the angle, 0, whose tangent. equals. Now tan, so lies in quadrant II. The calculator yields tan 0., which is an angle in quadrant IV. Since is in quadrant II, Therefore, cot.7. Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

18 Section 7.: The Inverse Trigonometric Functions [Continued] cot 8. tan 8. We seek the angle, 0, whose tangent. equals. Now tan, so lies in quadrant II. The calculator yields tan 0., which is an angle in 8. quadrant IV. Since is in quadrant II, Thus, cot ( 8.).0.. cot tan We are finding the angle, 0, whose tangent equals. Now tan, so lies in quadrant II. The calculator yields tan 0.9, which is an angle in quadrant IV. Since is in quadrant II, Thus, cot... csc sin We seek the angle,, 0, whose sine equals. Now, so lies in quadrant IV. The calculator yields sin 0.7, which is an angle in quadrant IV, so csc 0.7. cot 0 tan 0 We are finding the angle, 0, whose. tangent equals. Now tan, so 0 0 lies in quadrant II. The calculator yields tan 0.0, which is an angle in 0 quadrant IV. Since is in quadrant II, So, cot ( 0 ).8.. sec cos We are finding the angle, 0,, whose cosine equals. Now, so lies in quadrant II. The calculator yields cos., which is an angle in quadrant II, so sec.. 7. Let tan u so that tan u, < <, < u <. Then, cos ( tan u) sec sec + tan + u 8. Let cos u so that u, 0, u. Then, sin cos u sin cos u Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

19 Chapter 7: Analytic Trigonometry 9. Let sin u so that u,, u. Then, tan ( sin u) tan cos sin u u 0. Let cos u so that u, 0, u. Then, tan ( cos u) tan sin cos u u. Let sec u so that sec u, 0 and, u. Then, sin sec u sin cos sec sec sec u u. Let cot u so that cot u, 0 < <, < u <. Then, sin ( cot u) sin csc + cot + u. Let csc u so that csc u,, u. Then, ( u) cos csc cot cot cot csc csc csc csc u u. Let sec u so that sec u, 0 and, u. Then, cos( sec u) sec u. Let cot u so that cot u, 0 < <, < u <. Then, tan ( cot u) tan cot u. Let sec u so that sec u, 0 and, u. Then, tan sec u tan tan 7. sec u g f cos sin Let sin. Since and, is in quadrant I, and we let y and r. Solve for : ± ± Since is in quadrant I,. g f cossin r Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

20 Section 7.: The Inverse Trigonometric Functions [Continued] 8. f g sin cos Let cos. Since and 0, is in quadrant I, and we let and r. Solve for y: + y + y 9 y 7. h g tan cos Let cos. Since and 0, is in quadrant II, and we let and r. Solve for y: ( ) + y + y y y ± ± Since is in quadrant I, y. f g g f y sin cos r 7 7 f cos sin cos g sin cos sin 7. y ± 9 ± Since is in quadrant II, y. hg tan cos y tan g h cos tan Let tan. Since tan and, is in quadrant I, and we let and y. Solve for r: r + 7. h f tan sin Let sin. Since and, is in quadrant IV, and we let y and r. Solve for : + ( ) + 9 ± ± Since is in quadrant IV,. h f tan sin y tan 7. r + 9 r ± 9 ± Now, r must be positive, so r. gh costan r f h sin tan Let tan. Since tan and, is in quadrant I, and we let and y. Solve for r: r + r + 9 r ± 9 ± Now, r must be positive, so r. f h y sin tan r 7 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

21 Chapter 7: Analytic Trigonometry g g f cos sin cos f cos sin cos h g tan cos Let cos. Since and 0, is in quadrant II, and we let and r. Solve for y: ( ) + y + y y y ± Since is in quadrant II, y. hg tan cos y tan h f tan sin Let sin. Since and, is in quadrant IV, and we let y and r. Solve for : + ( ) + ± Since is in quadrant IV,. h f tan sin y tan 79. a. Since the diameter of the base is feet, we have r. feet. Thus,. cot.89. r b. cot h r cot r hcot h Here we have.89 and h 7 feet. Thus, r 7cot (.89 ) 7. feet and the diameter is ( 7.). feet. r c. From part (b), we get h. cot The radius is feet. r h 7.9 feet. cot. / Thus, the height is 7.9 feet. 80. a. Since the diameter of the base is.8 feet,.8 we have r. feet. Thus,. cot 0. r b. cot h r r cot h h cot Here we have 0. and r feet. Thus, h.79 feet. The cot ( 0. ) bunker will be.79 feet high.. c. TG cot.88 From part (a) we have USGA 0.. For steep bunkers, a larger angle of repose is required. Therefore, the Tour Grade 0/0 sand is better suited since it has a larger angle of repose. 8 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

22 Section 7.: Trigonometric Equations 8. a. b. cot y+ gt cot y+ gt The artillery shell begins at the origin and lands at the coordinates ( 7,0 ). Thus, 7 cot cot.788. The artilleryman used an angle of elevation of.. vt 0 sec sec v0 t 90. ft/sec 8. Let. y cot cos ( 7) sec(. ) y csc sin _ 0 0 _ 8 8. Answers will vary. Section The solution set is.., Note that the range of tan will not work. y sec cos y cot is 0,, so.. 0 ( )( ) or + 0 or The solution set is,. 0 ( ) () ± 0 ± + ± The solution set is +,. 9 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

23 Chapter 7: Analytic Trigonometry.. () () 0 [ ][ ] ( ) or 0 0 or The solution set is 0,. Let y and y. Use INTERSECT to find the solution(s): In this case, the graphs only intersect in one location, so the equation has only one solution. Rounding as directed, the solutions set is { 0.7 }... + k or + k, k is any integer On 0 <, the solution set is,. cos cos ± + k or + k, k is any integer On the interval 0 <, the solution set is,,, , + k, + k, k is any integer 9. False because of the circular nature of the functions. 0. False, is outside the range of the sin function k or + k, k is any integer 7 On 0 <, the solution set is,... tan tan ± ± + k or + k, k is any integer On the interval 0 <, the solution set is 7,,,. sin 0 sin sin ± ± + k or + k, k is any integer On the interval 0 <, the solution set is 7,,,. 70 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

24 Section 7.: Trigonometric Equations. cos 0 cos cos ± + k or + k, k is any integer On the interval 0 <, the solution set is 7,,,. sin + k k +, k is any integer On the interval 0 <, the solution set is 7,,. 7. ( ) 8. tan + k, k is any integer + k, k is any integer On 0 <, the solution set is. + k or + k + k or + k, k is any integer On the interval 0 <, the solution set is,,,. 9. cos ( ) 0. tan ( ) + k, k is any integer k +, k is any integer 8.. On the interval 0 <, the solution set is 7,,, sec + k or + k k 8 k + or +, 9 9 k is any integer On the interval 0 <, the solution set is 8,, cot + k, k is any integer k +, k is any integer On 0 <, the solution set is k or + k, k is any integer 7 On 0 <, the solution set is, k, k is any integer On the interval 0 <, the solution set is { }. tan + 0 tan + k, k is any integer 7 On 0 <, the solution set is,.. 7 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

25 Chapter 7: Analytic Trigonometry. cot + 0 cot cot + k, k is any integer On 0 <, the solution set is,. 7. sec + sec 8 sec + k or + k, k is any integer On 0 <, the solution set is,. 8. csc csc csc + k, k is any integer On 0 <, the solution set is k or + k, k is any integer On 0 <, the solution set is, k or + k, k is any integer On 0 <, the solution set is,.. + k + k + k, k is any integer 7 On 0 <, the solution set is,.. sin k 8 + k 9 k +, k is any integer 7 On the interval 0 <, the solution set is 0,, tan k + k + k, k is any integer On 0 <, the solution set is.. cos + k or + k 7 + k or + k 7 + k or + k, k is any integer. 7 On 0 <, the solution set is. 7 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

26 Section 7.: Trigonometric Equations. + kor + k, k is any integer. Si solutions are 7 9,,,,,.. tan k, k is any integer 9 7 Si solutions are,,,,,. tan + k, k is any integer Si solutions are 7 9,,,,,. 7 + k or + k, k is any integer. Si solutions are ,,,,, k or + k, k is any integer 7 9 Si solutions are,,,,,. + k or + k, k is any integer Si solutions are,,,,,. + k or + k, k is any integer. cos( ) + k or + k, k is any integer 7 8 Si solutions are,,,,,.. sin ( ). + k, k is any integer + k, k is any integer Si solutions are 7 9,,,,,. sin + k or + k, k is any integer k or + k, k is any integer. Si solutions are 8 0 0,,,,,.. tan + k, k is any integer + k, k is any integer Si solutions are 7 9,,,,,.. 0. sin ( 0.) or The solution set is { 0.,.7 }. 7 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

27 Chapter 7: Analytic Trigonometry cos ( 0.) or The solution set is { 0.9,. }. tan tan.7.7 or The solution set is {.7,. }.. tan tan 9 9 tan 9 tan or The solution set is {.08,. } cot tan tan or ,.. The solution set is { } 0.9 cos ( 0.9).9.9 or.9.9. The solution set is {.9,.9 }. 0. sin ,.08. or ( 0.0) The solution set is { } sec cos.8.8 or.8.. The solution set is {.8,. }. csc sin or ( 0.)..9.8 The solution set is {.8,.9 }..... cot cot tan tan or The solution set is {.7,. }. 0 sin or The solution set is { 0.7,. }. + 0 cos.. or..8. The solution set is {.,.8 }. 7 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

28 Section 7.: Trigonometric Equations cos + 0 cos ( + ) 0 0 or + 0,, The solution set is,,,. sin 0 ( + )( ) or 0 The solution set is,. sin 0 ( + )( ) or 0 7, 7 The solution set is,,.. (tan )(sec ) 0.. tan 0 or sec 0 tan sec, 0 The solution set is 0,,. (cot + ) csc 0 cot + 0 or csc 0 cot 7 csc, (not possible) 7 The solution set is,. sin cos + cos cos + cos + cos + 0 ( + ) 0 0 or + 0,, The solution set is,,,. 0. cos + 0 ( + )( ) or 0, The solution set is,,.. cos sin + 0 sin sin + 0 sin + 0 sin 0 ( + )( ) or 0 7, 7 The solution set is,,. 7 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

29 Chapter 7: Analytic Trigonometry. sin ( cos ( ) + ) sin ( cos + ) cos + cos ( )( ) cos + cos or + 0 (not possible) The solution set is { }.. sin ( cos ) sin ( ) ( ) cos cos cos cos cos + 0 ( )( ) cos cos 0 0 or 0 0, The solution set is 0,,. 7. sin tan, The solution set is,. 8. ( ) cos sin 0 cos sin tan 7, 7 The solution set is,. 9. tan 0 0 sin ( ) 0 or 0 0,, The solution set is 0,,,. 70. tan cot tan tan tan tan ± 7,,, 7 The solution set is,,,. 7 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

30 Section 7.: Trigonometric Equations cos + (sin ) + sin sin + 0 ( )(sin + ) 0 0 or + 0, The solution set is,,. sin + cos + cos ( ) cos The solution set is { } sin + 0 ( )( ) sin sin or 0 sin (not possible) The solution set is. cos 7 0 ( + )( ) or 0 (not possible), The solution set is, ( cos ) sin cos cos + 0 ( )( ) cos cos 0 0 or 0 0 (not possible) The solution set is { 0 }. ( + sin ) cos + sin sin ( )( ) sin + sin or + 0 (not possible) The solution set is. tan sec sec sec sec sec sec sec 0 (sec + )(sec ) 0 sec + 0 or sec 0 sec sec, (not possible) The solution set is,. csc cot + + cot cot + cot cot 0 cot (cot ) 0 cot 0 or cot,, The solution set is,,,. 77 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

31 Chapter 7: Analytic Trigonometry 79. sec + tan 0 tan + + tan 0 This equation is quadratic in tan. ` The discriminant is b ac < 0. The equation has no real solutions. 80. sec tan + cot + sin + cos Since sec and tan do not eist, the equation has no real solutions cos 0 Find the zeros (-intercepts) of Y + cos: , 0, sin Find the intersection of Y 7sin and Y : cos Find the intersection of Y 9+ 8cos and Y : ,.98,.8 8. sin 0 Find the zeros (-intercepts) of Y sin : sin + cos Find the intersection of Y sin + cos and Y :. 78 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

32 Section 7.: Trigonometric Equations 8. sin cos Find the intersection of Y sin cos and Y : cos Find the intersection of Y + cos : 0 Y and cos 0 Find the zeros (-intercepts) of Y cos:.0,.0 + sin 0 Find the zeros (-intercepts) of Y + sin : 0., sin e, > 0 Find the intersection of Y sin e and Y : 0 0.7, cos e, > 0 Find the intersection of Y cos e and Y :.7, sin Find the intersection of Y Y : 0,. sin and f 0 sin 0 sin sin sin ± ± + k or + k, k is any integer 79 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

33 Chapter 7: Analytic Trigonometry On the interval [ 0, ], the zeros of f are,,,. 9. f 0 cos + 0 cos cos + k or + k k k + or +, 9 9 k is any integer On the interval [ 0, ], the zeros of f are 8,, f sin 0 sin 0 0+ k or + k, k is any integer,, the zeros of f are 9. a. 0 On the interval [ ],,0,,,,. On the interval [, ], the solution set is 7 7,,,,,. d. From the graph in part (b) and the results of part (c), the solutions of f ( ) > on the 7 interval [,] is < < 7 or < < or < <. f cos 0 cos 0 + k or + k, k is any integer,, the zeros of f are 9. a. 0 On the interval [ ] 7,,,,,. b. f cos b. f sin c. f sin sin + k or + k, k is any integer f c. cos cos 7 + k or + k, k is any integer On the interval [,], the solution set is ,,,,,. d. From the graph in part (b) and the results of part (c), the solutions of f ( ) < on the 80 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

34 Section 7.: Trigonometric Equations 7 is < < or < < or < <. interval [, ] 97. f tan f tan tan + k, kis any integer a. f tan < tan < Graphing y tan and y on the interval,, we see that y < y for < < or,. b. < _ 98. f cot a. _ f cot + k, kis any integer f > cot > Graphing y tan and y on the interval ( 0, ), we see that y > y for 0 < < or 0,. b a, d. f sin + ; b. f g g 7 7 sin + sin sin + k or + k + k or + k, k is any integer On [ 0, ], the solution set is,. c. From the graph in part (a) and the results of f > g on part (b), the solution of [ 0, ] is < < or,. g 00. a, d. f ( ) cos + ; 8 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

35 Chapter 7: Analytic Trigonometry b. f g cos + cos cos + k or + k 0 + k or + k, k is any integer 0 On [ 0, ], the solution set is,. c. From the graph in part (a) and the results of f < g on part (b), the solution of [ 0, ] is 0 < < or 0,. 0. a, d. f cos ; g cos+ b. f g cos cos+ cos cos + k or + k, k is any integer On [ 0, ], the solution set is,. c. From the graph in part (a) and the results of f > g on part (b), the solution of [ 0, ] is < < or,. 0. a, d. f sin ; g sin + b. f g sin sin+ sin sin + k or + k, k is any integer On [ 0, ], the solution set is,. c. From the graph in part (a) and the results of f > g on part (b), the solution of [ 0, ] is < < or, P() t sin t a. Solve () sin t sin t 0 7 sin t 0 7 t k, k is any integer t k, k is any integer 7 P t on the interval [ ] We need 0 7 k, or 0 k 7. 0,. 8 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

36 Section 7.: Trigonometric Equations For k 0, t 0 sec. For k, t 0. sec. 7 For k, t 0.8 sec. 7 The blood pressure will be 00 mmhg after 0 seconds, 0. seconds, and 0.8 seconds. b. Solve P() t 0 on the interval [ 0, ] sin t 0 7 0sin t 0 7 sin t 7 t k +, k is any integer ( k + ) t, k is any integer 7 We need ( k + ) k + 7 k k For k 0, t 0. sec The blood pressure will be 0mmHg after 0. sec. c. Solve P() t 0 on the interval [ 0, ] sin t 0 7 0sin t 7 sin t 7 t sin t sin 7 On the interval [ 0, ], we get t 0.0 seconds, t 0.9 seconds, and t 0.89 seconds. Using this information, along with the results from part (a), the blood pressure will be between 00 mmhg and 0 mmhg for values of t (in seconds) in the interval 0, ,0. 0.8,0.89. [ ] [ ] [ ] 0. ht () sin 0.7t + a. Solve ht () sin 0.7t + on the interval [ 0, 0 ]. sin 0.7t + sin 0.7t 0 sin 0.7t t k, k is any integer 0.7 t k +, k is any integer k + t, k is any integer For k 0, t 0 seconds For k, t 0 seconds For k, t 0 seconds. 0.7 So during the first 0 seconds, an individual on the Ferris Wheel is eactly feet above the ground when t 0 seconds and again when t 0 seconds. b. Solve ht () sin 0.7t + 0 on the interval [ 0,80 ]. sin 0.7t + 0 sin 0.7t sin 0.7t 8 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

37 Chapter 7: Analytic Trigonometry 0.7t + k, k is any integer 0.7t + k, k is any integer + k t, k is any integer 0.7 For k 0, t 0 seconds For k, t 0 seconds For k, t 00 seconds. 0.7 So during the first 80 seconds, an individual on the Ferris Wheel is eactly 0 feet above the ground when t 0 seconds and again when t 0 seconds. c. Solve ht () sin 0.7t + > on the interval [ 0, 0 ]. sin 0.7t + > sin 0.7t > 0 sin 0.7t > 0 Graphing y sin 0.7 and y 0 on the interval [ 0, 0 ], we see that y > y for 0 < < So during the first 0 seconds, an individual on the Ferris Wheel is more than feet above the ground for times between about 0 and 0 seconds. That is, on the interval 0 < < 0, or ( 0,0 ). 70sin d a. ( ) 70sin ( 0) + 0 d 0 70sin miles 70sin on b. Solve d the interval [ 0, 0 ]. 70sin ( 0.) sin ( 0.) 0 sin ( 0.) 7 0. sin + k 7 sin + k k.9 + k or, k is any integer For k 0, or min 8. min For k, or min 8. min For k, or min 7.78 min So during the first 0 minutes in the holding pattern, the plane is eactly 00 miles from the airport when.0 minutes, 8. minutes,.7 minutes, and 8. minutes. 70sin > 00 on c. Solve d the interval [ 0, 0 ]. 70sin ( 0.) + 0 > 00 70sin ( 0.) >0 sin 0. > 7 and y on 7 Graphing y sin ( 0.) the interval [ ] 0, 0, we see that y > y for 0< <.0, 8. < <.7, and 8 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

38 Section 7.: Trigonometric Equations 8. < < So during the first 0 minutes in the holding pattern, the plane is more than 00 miles from the airport before.0 minutes, between 8. and.7 minutes, and after 8. minutes. d. No, the plane is never within 70 miles of the airport while in the holding pattern. The minimum value of sin ( 0. ) is. Thus, the least distance that the plane is from the miles. airport is 0 0. R 7sin a. Solve R 7sin 0 on the interval 0,. 7sin ( ) 0 0 sin ( ) 7 sin + k sin + k 0.77 k.08 k + or +, k is any integer For k 0, For k, or or So the golfer should hit the ball at an angle of either.0 or b. Solve R 7sin 0 on the interval 0,. 7sin ( ) 0 0 sin ( ) 7 8 sin + k 8 sin + k k.08 k + or +, k is any integer For k 0, or For k, or So the golfer should hit the ball at an angle of either.7 or.. c. Solve R 7sin 80 on the interval 0,. 7sin ( ) sin ( ) 7 ( ) sin 7 Graphing y sin and y on the 7 interval 0, and using INTERSECT, we see that y y when radians, or _ 8 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall.

39 Chapter 7: Analytic Trigonometry 0. _ c. Graph Y cos + sin and use the MINIMUM feature: 0. So, the golf ball will travel at least 80 feet if the angle is between about.79 and 7.. d. No; since the maimum value of the sine function is, the farthest the golfer can hit the ball is 7() 7 feet. 07. Find the first two positive intersection points of Y and Y tan. 0 0 The first two positive solutions are.0 and a. Let L be the length of the ladder with and y being the lengths of the two parts in each hallway. L + y y y L ( ) sec csc + + sec tan csccot 0 sec tan csccot sec tan csc cot tan tan º 09. a An angle of 7.7 minimizes the length at L 9.87 feet. d. For this problem, only one minimum length eists. This minimum length is 9.87 feet, and it occurs when 7.7. No matter if we find the minimum algebraically (using calculus) or graphically, the minimum will be the same. ( ) (.8) sin (9.8) sin ( ) 0.89 (.8) sin ( 0.89) 0º or 0º 0º or 0º b. Notice that the answers to part (a) add up to 90. The maimum distance will occur when the angle of elevation is 90 : ( ) (.8) sin R( ). 9.8 The maimum distance is. meters. (.8) sin c. Let Y 9.8 d L + cos 7.7º sin 7.7º b. ( 7.7º ) 9.87 feet 8 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

40 Section 7.: Trigonometric Equations 0. a. (0) sin( ) sin( ) sin ( 0.77).º or 7.º.º or 8.8º b. The maimum distance will occur when the angle of elevation is : (0) sin[ ( )] R( ). 9.8 The maimum distance is approimately. meter (0) sin c. Let Y : d sin 0.. sin 0 sin sin Calculate the inde of refraction for each: v v sin0º 0º 8º.77 sin 8º sin 0º 0º º 0'.º.798 sin.º sin 0º 0º º0'.º.0 sin.º sin 0º 0º 9º 0' 9º.9 sin 9º sin 0º 0º º 0' º. sin º sin 0º 0º 0º0' 0.º. sin 0.º sin 70º 70º º0'.º.7 sin.º sin80º 80º 0º 0' 0º.8 sin 0º. Yes, these data values agree with Snell s Law. The results vary from about. to.. 8 v v.9 0 The inde of refraction for this liquid is about... Calculate the inde of refraction: sin 0º 0º, º;.7 sin º. sin 0.. sin 0 sin sin The inde of refraction of crown glass is.. sin 0º.. sin 0 sin sin The angle of refraction is about Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

41 Chapter 7: Analytic Trigonometry 7. If is the original angle of incidence and φ is the angle of refraction, then n sinφ. The angle of incidence of the emerging beam is also φ, and the inde of refraction is. Thus, is n the angle of refraction of the emerging beam. The two beams are parallel since the original angle of incidence and the angle of refraction of the emerging beam are equal. 8. Here we have n. and n.. nb ncos B B n B n n tanb n n. B tan tan 8.8 n. 9. Answers will vary. 0. Since the range of y sin is y, then y sin + cannot be equal to when > or < since you are multiplying the result by and adding. Section 7.. True. True. identity; conditional 7. False, you need to work with one side only. 8. True tan csc cot sec ( + ) + sin sin + sin sin sin cos + ( + ) ( ) cos + cos cos cos cos sin ( ) + + sin + + ( ) sin + + cos sin + cos + + cosv+ cosv + cosv + cosv cosv + cosv cos v sin v.. 0. True 88 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

42 Section 7.: Trigonometric Identities ( )( ) sin + cos sin + cos sin + + cos sin + cos tan tan sec tan tan + tan + sec tan tan + + tan sec tan sec + tan sec tan tan tan sin sin + + cos ( + )( + ) ( )( ) cos + + ( + )( ) ( ) cos (tan + cot ) + sin + cos csc sin (cot + tan ) + cos + sin sec tan cot cos tan cos tan u cos u u u u u u sin u sin csc cos sin cos sin u cos u u u u u u sin u (sec )(sec ) sec tan + (csc )(csc ) csc cot + 9. csc cot 9. (sec + tan )(sec tan ) sec tan 0... sec tan + tan ( ) + ( tan ) + tan sec + cot ( ) + ( cot ) + cot csc 0.. (csc + cot )(csc cot ) csc cot cos ( + tan ) cos sec cos cos 89 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

43 Chapter 7: Analytic Trigonometry.. ( cos )( + cot ) sin csc sin sin ( + cos ) + ( cos ) sin + + cos + sin + cos sin + cos (sin + cos ) cosu cscu cot u sin u sin u cosu + cosu sin u + cosu cos u sin u( + cos u) sin u sin u( + cos u) sin u + cosu sin + cos sin + cos + cos (sin + cos ) + cos tan cos + cot sin sin cos + cos sin sin + cos cos sin sec sec sec (sec ) (tan + ) tan tan + tan csc csc csc (csc ) (cot + )cot cot + cot sinu secu tan u cosu cosu sin u + sin u cosu + sin u sin u cos u( + sin u) cos u cos u( + sin u) cosu + sinu cos + cos 9sec tan sec + sec tan sec + (sec tan ) sec + + sec cos sin + + ( sin )(+ sin ) + ( ) + sin cos ( cos )( + cos ) (+ cos ) + + tanv cot v tanv cot v + cotv cot v cotv cot v cot v + cot v 90 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

44 Section 7.: Trigonometric Identities. cscv sin v cscv + + sin v sinv sin v + sinv sin v sinv + sinv sec sec + sec sec sec sec + sec sec.. 7. sec cos + + csc + tan + tan tan csc csc csc + cot cot csc + csc cot (csc + ) cot cot (csc + ) cot csc csc csc csc + csc csc csc csc + csc csc csc csc + csc sin cos ( sin ) + cos + cos v sin v cos v( sin v) v v v v sin + sin + cos cos v( sin v) v v v sinv + cos v( sin v) sinv cos v( sin v) ( sin v) cos v( sin v) cos v secv cosv + sin v cos v+ ( + sin v) + + sinv cosv cos v(+ sin v) cos v+ + sin v+ sin v cos v( + sin v) + sinv cos v( + sin v) ( + sin v) cos v( + sin v) cos v secv sin cot 9 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

45 Chapter 7: Analytic Trigonometry... sin cos + + (sec tan ) ( )( + ) + cos ( ) sec sec tan + tan sin + cos cos cos cos + sin cos ( sin )( sin ) sin (sin )(sin ) ( sin )( + sin ) + (csc cot ) csc csc cot + cot cos + sin sin sin sin + cos sin ( cos )(cos ) cos (cos )( cos ) ( cos )(+ cos ) tan cot + + cos sin + cos sin ( sin )( + sin ) + cot tan + tan cot + + cos sin + sin ( sin ) cos ( cos ) cos cos sin sin + cos ( cos ) sin cos cos ( cos ) ( cos )(sin cos ) + + cos ( cos ) sin + + cos sin cos tan + cot 9 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

46 Section 7.: Trigonometric Identities tan sin ( + sin ) + cos cos ( + sin ) sin + sin + cos cos ( + sin ) + cos ( + sin ) sec (cos ) cos cos sin (cos sin ) cos sin cos tan tan tan + sec tan sec + tan + (sec ) tan + (sec ) tan (sec ) tan + (sec ) tan + tan (sec ) + sec sec + + tan (sec sec ) sec sec + sec sec sec + tan (sec ) sec sec (sec ) + tan (sec ) sec (sec )(sec + tan ) (sec ) sec tan (sec ) sec sec tan + sec ( cos ) + ( + cos ) + ( + cos ) ( + cos ) + sin cos ( + cos ) sin cos sin cos sin ( sin ) sin + sin ( + ) + tan cot tan + cot + sin cos sin + cos sin cos sin cos cos sec sec + cos cos + cos + cos cos + cos sin + cos 9 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

47 Chapter 7: Analytic Trigonometry... sin u cosu tan u cot u + cosu sin u + tan u+ cot u sin u cosu + cosu sin u sin u cos u cosusin u + sin u+ cos u cosusin u sin u cos u + sin u cos u+ u+ u sin ( cos ) sin u+ sin u sin u sin u cosu tan u cot u cos sin + cos u u u + cos u tan u+ cot u sin u cosu + cosu sin u sin u cos u cosusin u + cos u sin u+ cos u cosusin u sin u cos u + cos u sin u+ cos u + sec + tan cot cos ( + sin ) tansec sec + sec + cos + + cos sin tan tan + tan tan + tan + tan tan + + tan + tan + tan sec sec cos cot cot + cos + cos + cot csc cot + cos csc csc cos sin sin + cos sin sin cos + cos sin + cos sec csc sec csc seccsc seccsc seccsc csc sec 9 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

48 Section 7.: Trigonometric Identities sin tan cos cot sin cos sin cos sin cos sin cos ( ) sin cos tan sec cos sin tan tan + cot + sin + cos seccsc ( + sin ) ( sin ) ( sin )(+ sin ) + + sin ( + sin ) sin cos cos cos tansec sec sec + + sec ( + sin ) sin sec ( + sin ) cos + cos + cos + sin (+ sin )(+ sin ) sin ( sin )(+ sin ) ( + sin ) sin ( + sin ) cos + + (sec + tan ) sin ( sin )( + sin ) sin cos sec 9 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

49 Chapter 7: Analytic Trigonometry (secv tan v) + csc v(sec v tan v) sec v secvtan v+ tan v+ csc v(secv tan v) sec v sec vtan v+ sec v csc v(secv tan v) sec v secvtanv csc v(sec v tan v) sec v(sec v tan v) csc v(sec v tan v) secv cscv cos v sin v sinv cos v sin v cos v tanv v v+ v + v secv secv sin v + cos v cos v cos v+ sin v cos v cos v sec tan tan tan cos v+ sin v sin + cos seccsc cos + sin seccsc sin + cos + + sin + cos (sin + cos )(sin sin cos + cos ) sin + cos cos ( + cos )(sin + cos ) cos cos ( + cos )(sin + cos cos ) sin cos ( + cos )( cos ) ( + cos )( cos ) sec tan cos sin cos sin tan sin cos cos sin cos sin cos ( ) cos cos sin cos sin cos 9 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

50 Section 7.: Trigonometric Identities cos sin sin cos sin sin + cot + sin + cot + cos (cos ) cos (sin + cos ) + cos sin (cos sin )(cos sin ) (cos sin ) (cos sin )(cos sin ) + cos sin cos + sin cos sin sin sin sin cos cos cos sin cos sin cos tan cot ( + sin ) + cos ( + sin ) + (+ sin ) cos ( + sin ) sin + cos ( + sin ) + cos + + sin cos + + sin + cos ( + sin ) + ( sin ) sin ( sin ) cos (+ sin ) + sin ( + sin ) + cos ( + sin ) sin ( + sin ) ( + sin )( + cos ) sin (+ sin ) ( + cos ) + sin ( + cos ) + (+ cos ) sin ( + cos ) + + cos + cos + sin (+ cos ) + sin + + cos sin + + cos (cos ) + cos + cos + sin (+ cos ) + cos + + sin (+ cos ) + cos ( + cos ) + sin ( + cos ) cos (+ cos ) ( + cos )( + sin ) cos (+ cos ) + + sec + tan ( a + bcos ) + ( a bsin ) a sin + ab + b cos + a cos ab + b sin a (sin + cos ) + b (sin + cos ) a + b (acos ) + a (cos sin ) a sin cos + a ( cos cos sin + sin ) ( sin cos cos cos sin sin ) ( cos cos sin sin ) ( cos sin ) a + + a + + a + a a () 97 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

51 Chapter 7: Analytic Trigonometry tanα + tan tanα + tan 9. cotα + cot + tanα tan tanα + tan tan + tanα tanα tan tanα tan (tanα + tan ) tanα + tan tanα tan 9. (tanα + tan )( cotαcot ) + (cotα + cot )( tanα tan ) tanα + tan tanαcotαcot tan cotαcot + cotα + cot cotα tanα tan cot tanα tan tanα + tan cot cotα + cotα + cot tan tanα (sinα + cos ) + (cos + sin α)(cos sin α ) sin α + sinαcos + cos + cos sin α sin cos + cos α cos (sinα + cos ) (sinα cos ) + (cos + sin α)(cos sin α) sin α sinαcos + cos + cos sin α α + α sin cos cos cos (sin cos ) ln sec ln ln ln ln tan ln ln ln 98. ln sec + tan + ln sec tan ln sec + tan sec tan ln 0 ( ) ln sec tan ln tan + tan 99. f sin tan sin sin cos sin cos cos cos cos cos cos sec cos g 00. f cos cot cos cos sin cos sin sin sin sin sin sin csc sin g 97. ln + + ln ( ) ln + cos cos ln cos ln sin ln 98 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

52 Section 7.: Sum and Difference Formulas 0. f ( ) + ( )( + ) ( + ) ( + ) sin cos ( + ) ( sin + cos ) ( + ) ( + ) 0 ( + ) 0 g 0. f tan + sec sin ( ) cos ( ) g ( ) 00sec sec 00 cos cos 00 cos cos cos 00 cos cos cos 00( + cos ) cos 00( + sin ) cos 0. ( ) 0. It ( csc )( sec + tan) A cscsec csc sec + tan A csc sec tan A + csc sec A + ( ) A sin ( A ) A cos cos 0. Answers will vary. 0. sin + cos tan + sec + cot csc Answers will vary. Section 7.. ( ) + ( ).. a. + + b. 9. y, r, (Quadrant ) cosα r.. 7. False 8. False 9. False 0. True 99 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

53 Chapter 7: Analytic Trigonometry.... sin sin + sin cos + cos sin + ( + ) sin sin sin cos cos sin ( ) 7 cos cos + cos cos sin sin ( ) 7 tan tan + tan + tan tan tan cosº cos( 0º + º ) cos0º cos º sin0º sin º ( + ). sin0º sin ( 0º + º ) sin 0º cos º + cos0º sin º + ( + ) 7. tanº tan(º 0º ) tan º tan 0º + tan º tan 0º tan9º tan(º + 0º ) tanº + tan 0º tanº tan 0º + ( ) Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

54 Section 7.: Sum and Difference Formulas sin sin + sin cos + cos sin + ( + ) 9 tan tan + tan + tan tan tan cot cot tan tan + tan + tan tan tan tan tan tan + tan sec cos cos cos cos + sin sin sin 0º cos0º + cos0º sin0º sin(0º + 0º ) sin 0º. sin 0º cos80º cos0º sin80º sin(0º 80º ) sin( 0º ) sin 0º. cos70º cos 0º sin 70º sin 0º cos(70º + 0º) cos 90º 0. cos0º cos0º + sin 0º sin0º cos(0º 0º ) cos 0º 70 Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

55 Chapter 7: Analytic Trigonometry tan0º + tanº tan0ºtanº tan º 7. tan ( 0º + º ) tan 0º tan0º + tan 0º tan0º tan 0º 8. tan ( 0º 0º ). sin α, 0 < α < cos, < < 0 y α (, ) y y (, y) sin cos cos sin sin sin sin cos cos sin sin cos + cos cos cos cos + sin sin cos cos cos cos sin cos + cos sin sin sin 8 sin +, > 0 9, > 0 cos α, tanα y +, y < 0 y 0, y < 0 y sin, tan a. sin( α + ) sinα cos + cosα sin + b. cos( α + ) cosα cos sinα sin Copyright 0 Pearson Education, Inc. Publishing as Prentice Hall

Areas and Lengths in Polar Coordinates

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