MATH 1316 REVIEW FOR FINAL EXAM

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1 MATH 116 REVIEW FOR FINAL EXAM Problem Answer 1. Find the complete solution (to the nearest tenth) if 4.5, 4.9 sinθ and 0 θ < π.. Solve sin θ 0, if 0 θ < π. π π,. How many solutions does cos θ cosθ have if 1 0 θ < π? 4. Solve tan θ tanθ 0 for 0 θ < π (in radians). π 7π,, 1.49, Solve tan θ tanθ 0 for 0 θ < π. π 5π 0,,, π Solve cos θ 1 cosθ for 0 θ < π. 1.86, Solve sin θ cosθ for 0 θ < π. 1.6, Solve sin θ+ cosθ 1 0 for 0 θ < 60. 0, 10, Solve tan θ+ secθ 1 for 0 θ < 60. 0, 10, Solve sinθ+ 1 cos θ for 0 θ < 60. Round θ to 16, 164 the nearest whole degree. 11. Solve cos θ sinθ for 0 θ < π in radians correct to 0.08,.84 three decimal places. 1. Find the sum of the solutions of cos θ sinθ for θ < Solve cosθ sin θ for 0 θ < π. π 5π π π Find the sum of the solutions of 5tanθ 5 0 if θ < Evaluate (in radians): A. Sin B. Cos 1. 4 C. Arccos. D. Arcsin 1 A B C. not defined D //001

2 16. Find the given angle (in radians) from memory. A. Cos 1 1 B. Arccos( 1 ). Find a positive coterminal angle less than one revolution 10//001 A. π B. π C. Arcsin C. π D. Sin D. π Give the exact values for A. Cot 1 A. π B. Arcsin B. π C. Arccos 1 C. π D. Cot ( ) 1 D. 5 π 6 E. cos150 F. cot E. π F. 0 π 18. Arccot Give the domain and range for y Arc sin x. Domain: 1 x 1 π π Range: y 0. State the quadrant where each angle terminates. A. -10 A. II B. 15 B. II C. 40 C. I D. -00 D. I E. 10 E. III F. -60 F. IV G. IV G. 15 H. II H Change the given measurements to decimal degrees. A B In which quadrant is an angle of 5? IV A B. 0.00

3 for each of the given angles. A. -0 B. 700 C D In what quadrant does 16 π terminate? 7 5. Change the degree measures to radian measures. A. 40 B. -90 C A. 0 B. 40 C. 60 D. 0 IV A. 4 π B. π C. 10 π 6. Change π 15 to degrees Find the length of an arc with central angle of 40 and π radius How far (to the nearest cm) does the tip of a pendulum 7 cm move if it is 5 cm long and swings through an arc of If P(s,t) is a point on the terminal side of an angle δ s which is also on a unit circle, the cosδ 0. If P(s,t) is a point on the terminal side of an angle δ which is on a unit circle, then tanδ 1. If β is an angle in standard position, what are the coordinates of the point of intersection of the terminal side of β and the unit circle. t s cos β,sin ( β ). What is the reciprocal of the sine? cosecant. What is the reciprocal of the secant? cosine 4. Write the Pythagorean Identities. sin θ+ cos θ 1 1+ tan θ sec θ cot θ+ 1 csc θ 5. What is the value of tan 14 sec 14? Find cotθ if secθ 5 and sinθ > In Quadrant I, which trigonometric functions are positive? All are positive in 8. In Quadrant II, which trigonometric functions are negative? Quadrant I. cosine, tangent, cotangent, secant 10//001

4 9. Which one of the following is NOT a fundamental identity? 1 A. sinθ cscθ B. sin θ 1 cos θ C. 1+ tan θ sec θ sinθ D. tanθ cosθ E. All are fundamental identities. 40. Which of the following is true? A. cos θ 1 cos6θ B. csc 5θ 1 cot 5θ C. 1 + cos 8 θ cos 4θ D. sinθcosθ sinθ E. all of the above 41. Which of the following statements is false? A. cos θ+ sin θ 1 B. tan θ sec θ 1 C. cosθsecθ 1 D. cot θ 1 csc θ E. None of these are false. 4. Which of the following is an equivalent form of sin tan θ θ cos θ? A. tanθcosθ sinθ sinθ B. cosθ tanθ C. tanθ sinθsecθ D. all of these 4. Which of the following is NOT an equivalent form of 1+ tan θ sec θ? A. tan θ sec θ 1 B. tanθ ± sec θ 1 C. secθ ± 1+ tan θ D. sec θ tan θ 1 E. All are fundamental identities. E. all of the above D. cot θ 1 csc D. all of these B. tanθ ± sec θ 1 θ 10//001

5 44. If sinθ 1 cos θ, then θ could be in quadrants A. I and II B. III and IV C. I and IV D. II and III 45. If cosθ 1 sin θ, then θ could be in quadrants A. I and II B. III and IV C. I and IV D. II and III 46. State the quadrants in which θ may lie to make the given expressions true. A. cos θ 1 sin θ and tanθ < 0 B. tanθ sec θ 1 and cosθ < 0 C. secθ 1+ tan θ and sinθ > 0 D. sinθ 1 cos θ and cosθ > Which of the following are true in Quadrant III? A. Sine and tangent are positive. B. Tangent and cosecant are positive. C. Cotangent is positive and cosine is negative. D. Cotangent and cosecant are negative. E. None of these are true. 48. Find the values of the trigonometric functions for an angle whose measure is 1.5. Round to four decimal places. 49. Give the reference angle for each of the following: A. 1 B. 18 C. 6 π 7 B. III and IV C. I and IV A. IV B. II C. I D. IV C. Cotangent is positive and cosine is negative. sin cos tan cot sec csc A. 51 B. 4 C. π 7 5π 50. Find the exact value of tan. 6 10//001

6 51. Using the definition of the trigonometric functions, find their values for an angle θ whose terminal side passes through (, ). 11π 5. Find the exact value of sin. π 5. Find the exact value of cos Cosecant is undefined for an angle of A. 0. B. π. C. π. D. all of these 55. If sin x 1 and π π x, the x? sinθ cosθ tanθ cotθ secθ cscθ D. all of these D. 5 π 6 A. π 6 B. π C. π D. 5 π Find the period of y+ cos( πx+ π ). 57. Find the amplitude of y cos x. π 58. For y+ 5 sin x+, find A. the amplitude. B. the period. 10//001 A. 1 B. π

7 59. For y tan 1 x+ 4 1, find A. the amplitude. B. the period. C. the phase shift. D. the vertical translation. 60. cot π A. 0 B. C. D. undefined π 61. The graph of y 4cos x+ is similar to y cos x but is shifted A. π units left. A. none B. π C. 8 left D. 1 down A. 0 C. π 6 units left B. π C. π 6 D. π 6 units right. units left units right 6. Simplify using only sines, cosines, and the fundamental identities: cos θ sinθ + sinθ secθ+ cscθ 6. Simplify using only sines and cosines. tanθcotθ 64. sec 7θ 1 is identical to A. tan θ B. cos 7θ C. tan7θ D. tan 7θ 1 sinθ sinθ+ cosθ sinθcosθ D. tan 7θ 10//001

8 65. ( secθ cosθ) is identical to A. sec θ cos θ B. sec θ+ cos θ C. tan θ cos θ D. tan θ sin θ 66. Verify the following identity: secθ 1 sinθ tanθ tanθsecθ 67. csc θ cosθ simplifies to cosθ sinθ A. sin θ B. cosθ C. cotθ D. tanθ ( tan θ+ 1) cos θ 68. If is changed to sines and cosines and cot θ then completely simplified, the result is 4 A. sin θ+ sin θ B. 1 C. cos 4 θ 4 sin θ sin θ D. cos θ 69. The expression sinθcosθ( tanθ+ cotθ) is identical to A. 1 B. cos θ C. sinθcosθ D. sinθ cosθ 4 4 sin θ cos θ 70. The best procedure for proving 1 is to sin θ 1 begin by A. breaking up the fraction. B. factoring. C. changing to sines and cosines. D. multiplying by the conjugate. D. tan θ sin θ Answers may vary. D. tanθ D. A. 1 sin cos θ θ B. factoring 10//001

9 71. Verify the following identity: tan β cot β+ cosβ+ 1 secβ sin β+ secβ ( ) ( ) Answers may vary. 7. The expression 1 + cosθ cos θ 1 sinθ can be D. cosθ+ sinθ 1 cos θ 1+ sinθ cos θ reduced to A. secθ+ tanθsecθ+ sec θ B. secθ+ 4 secθ sec θ C. secθ 1 tanθsecθ+ tan θ D. cosθ+ sinθ 1 cos θ 7. Which of the following is NOT an identity? D. tanθ tanθsecθ sinθ cosθ A. sinθ secθ tanθ B. secθ cotθ cos θ C. cotθsin θ cosθsinθ D. tanθsecθ sinθcosθ E. These are all identities. 74. Write cos1 in terms of its cofunction. sin Write sin( ) as a function of a positive acute angle. sin 76. Write cos( 9 ) as a function of a positive acute angle. cos9 77. The graph of y 6sin4θ has amplitude D. 6 A. π B. π 4 C. -6 D Evaluate sin10 cos185 cos10 sin185. sin Evaluate cos175 cos11 + sin175 sin11. cos Write ( ) tan 0 +θ as a function of θ only. 1+ tanθ tanθ π 81. Write cos θ as a function of θ only. 1 ( cosθ+ sinθ) 10//001

10 8. If α π and β θ and these values are substituted into ( ) cos α+ β cos αcosβ sinαsinβ, then the result is: π A. cos θ sin( θ ) π B. cos θ cos θ π C. cos θ sin( θ ) π D. cos θ sin θ 8. sin( θ+ A) + sin( θ A ) simplifies to A. sinθ B. sinθcosa C. cosθsina D. sinθcos A 84. If cos( α+ β) cos αcosβ sinαsinβ, then an exact value for cos165 is 6 A. 4 6 B C. 4 D Which of the following is NOT a double-angle identity? A. sinθ sinθcosθ B. cosθ cos θ sin θ C. cosθ 1 cos θ D. cosθ 1 sin θ tanθ E. tanθ 1 tan θ C. π cos θ sin( θ ) B. sinθcosa C. cosθ 1 cos θ 10//001

11 86. cos θ sin θ is identical to A. 1 B. tan θ C. cos 6θ D. cos 15θ. 87. If cosθ 5 and θ is in Quadrant IV, then sinθ is C. cos6θ A. 4 5 B C. 4 5 D If sinθ 5 and θ is in Quadrant III, find cos θ The exact value of cos15 is Which of the following is FALSE? E. They are all true cosθ A. cos θ ± 1 1 cosθ B. sin θ ± 1 1 cos θ C. tan θ sin θ 1 sinθ D. tan θ 1 + cosθ E. They are all true cosθ C. I or IV 91. If cos θ, then in which quadrants is the angle 1 θ? A. I or II B. I or III C. I or IV D. II or III 10//001

12 9. If cot θ, and θ is in the first quadrant, then tanθ is 4 5 A. 5 B. 5 5 C. 1 C. 1 D The expression sin4θ is equivalent to A. A. 4 sinθcos θ 4sin θcosθ 4sinθcos θ 4sin θcosθ B. 8sin θ 4sinθcosθ C. cos θ sinθcosθ sin θ D. cos θ sinθ cos θ C. cotθ simplifies to sinθ A. tanθ B. cotθ C. cotθ D. 1+ cot θ 95. Verify the following identity: Proofs vary. cotθsinθ cos θ sin θ. 96. Solve sin θ if 90 θ If 0 θ < π and 4 cos θ 4, then the complete B. π 11, π solution is 6 6 A. π 5, π 6 6 B. π 11, π 6 6 C. π 5, π D. π, π 98. Solve cos θ cosθ 0 for 0 θ < π in radians π (exact values). 99. Solve 4sin θ 1 0 for 0 θ < π. The sum of the D. 4π 10//001

13 solutions is A. π B. π C. π D. 4π 100. Solve tan θ+ secθ 1 for 0 θ < 60. 0, 10, Solve cos θ sinθ for 0 θ < 60. 0, 150, Give the exact values for A. Arc tan1 B. Cos 1 A. 45 or π 0 4 C. sin60 B. 90 or π C. 10. Evaluate (in radians) A. Sin B. Cos 1. 4 C. Arccos. D. Arc sin Find the given angle (exact value in radians) from memory. A. Sin 1 B. Arccos0 C. Cos 1 D. Arc sin 1 A B C. undefined D A. π 4 B. π C. π 6 D. π Find the given angle (in radians) from memory. A. Cos //001 B. Arc cos( 1 ) C. Arcsin D. Sin Simplify using exact values (in radians). A. Tan 1 1 A. π B. π C. π D. π

14 B. Cot ( ) 1 C. Sec 1 1 D. Arc csc( ) A. π 4 B. 5 π 6 C Give the exact values for A. csc 5 π 4 B. cos( 10 ) C. Cot The inverse tangent is defined in quadrants A. I and II. B. I and IV. C. III and IV. D. I and III What is the measure of the third angle of a triangle whose other angles are 1 and 10? 110. If x is negative in θ Arc cos x, then θ is in quadrant(s) A. I B. II C. III D. II and III D. π 6 A. B. C. 90 or π B. I and IV. 9 B. II 111. The angle of depression is the angle between the horizontal and the line of sight. 11. The angle of elevation of the Vehicle Assembly Building at the Kennedy Space Center is.8 at 150 feet from the base of the building. What is the height of this building which is used to house the space shuttle? 11. The Law of Cosines can be used if the given information is A. SAA. B. AAA. C. SSS. D. SSA.. 55 ft C. SSS. 10//001

15 114. The Law of Cosines can be used if the given D. SAS. information is A. AAA. B. ASA. C. SSA. D. SAS State the Law of Cosines. c a + b abcos C 116. The Law of Cosines for a correctly labeled triangle states A. a A. b + c bccos A B. cosβ B. a c b + ac 117. Solve a triangle whose sides are a5, b5, and c50. α8, β 41, γ If a14.6 in., b8. in., and c19.4 in., find angle β Solve a triangle with sides b5., c4.9, and angle C If a41 ft, b76 ft, and angle B109, side c is A. 50 ft. B. 5 ft. C. 54 ft. D. 48 ft If a5 cm, b48 cm, and angle A46, how many triangles are formed? 1. If a8 inches, b7 inches, and angle A4, angle B is A. 118 B. 6 C. 6 and 118 D Kristin, in the control tower of an airport, observes Brian in a DC9 plane at a distance of 50 km on a bearing of 100 and Judy in a 747 plane at a distance of 60 km on a bearing of 110. How far apart are Brian and Judy? 14. Solve a triangle with angles B11, C5, and side b94. In 1, a4.0, A46, B71 In, a.77, A8, B109 B. 5 ft. triangles C. 6 and km a55, c58, A 10//001

16 15. A surveyor needs to measure the distance across a 8 m swamp, so two points P and Q are selected at opposite ends of the swamp. From point P the surveyor determines that point Q is in the direction of N47 E. Next, the surveyor measures 100 m in an easterly direction and arrives at a point, call it R. The bearing of point Q from R is N5 W. How far is it across the swamp? 16. Find the area of a triangle with A48.6, c451, and 66,800 b Find the area of a triangle with sides 15, 18, and Find the area of a sector of a circle of radius 68 ft 770 ft subtended by an angle of What is the smallest force necessary to keep a 800 lb 90 car from sliding down a hill that makes an angle of 6.0 with the horizontal? Assume that friction is ignored. 10. Simplify: (-4i)+(-6+i) --i 11. Simplify: A. (-11+7i)-(8-i) B. (4+9i)+(-6+i) 1. The absolute value of 6-5i is A. 1 B. 6 C. 61 D Change to trigonometric form: A. -10 B. i 14. To change a complex number rcisθ to rectangular form use a and b. A i B. -+1i D. 61 A. 10cis180 B. 6cis40 rcosθ ; rsinθ 15. Change to rectangular form: A. cis60 7π 7π B. cos + i sin 6 6 A. + i B. i 10//001

17 ( 5 5i ) ( i) simplified is 1 i A. 5 5i B. 10( i ) C. 0cis0 D. 51 ( i) ( i ). cis18 9cis85 is ( cis4 ) A. 6cis409 B. 6cis49 C. 6cis401 D. 6cis Compute ( 1+ i ) 10. Express the answer in rectangular form. A. - B. -8i C. -i D. 8i 19. Find all the third roots of 7i. Leave the answers in trigonometric form Find the linear velocity of a point on the edge of a flywheel of radius 7 meters if the flywheel is rotating 90 times per second sin( θ ) is equivalent to A. cos( θ ) B. cosθ C. sinθ D. all of the above Two forces of 14 and 15 newtons act on a body at an angle of 11. Find the magnitude of the resultant force. 14. Fill in the blanks. A. tan18 7 B. cos sin55 A. 5 5i. -i cis0, cis150, cis70 160π meters second C. sinθ 09 newtons A. cot B. 5 10//001

18 144. Solve for x: sec ( 5x+ ) csc( x 9) Graph y + sin x for one period. 16 x 146. Graph y tan + π over a two-period interval. π 147. Graph y csc x + 1 for one period. 10//001