Math 1316 General Review for Trigonometry Last Updated 08/15/2014

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1 Math 11 General Review for Trigonometry Last Updated 08/1/01 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the measure of each angle in the problem. 1. Supplementary angles with measures x + 8 and x - degrees 8 and 9 98 and 8 18 and 11 and. Complementary angles with measures x and x - 9 degrees 8 and 9 and 11 and 9 and. x + y = 0, x 0; Find csc. 1 Evaluate the expression. 8. sin Undefined. A wheel makes 9 revolutions per minute. How many revolutions does it make per second? 1. revolutions per second revolutions per second.9 revolutions per second. revolutions per second. A wheel is rotating 0 times per minute. Through how many degrees does a point on the edge of the wheel move in 1 seconds? cos(-90 ) 0-1 Undefined Use the appropriate identity to find the indicated function value. Rationalize the denominator, if applicable. If the given value is a decimal, round your answer to three decimal places. 10. cot, given that tan = Suppose that is in standard position and the given point is on the terminal side of. Give the exact value of the indicated trig function for.. (18, ); Find csc. 11. tan, given that cot = An equation of the terminal side of an angle in standard position is given along with a restriction on x. Find the indicated trigonometric function value of. Do not use a calculator.. -x + y = 0, x 0; Find sin Use the fundamental identities to find the value of the trigonometric function. 1. Find tan, given that sin = and is in quadrant II

2 1. Find sec, given that tan = and is in 1. Find the exact value of x in the figure. quadrant I Find sec, given that tan = 0.0 and is in quadrant I Find tan, given that cos = and is in quadrant II x Find the exact value of x in the figure. Find the exact value of the expression. 18. cos sec cos (-190 ) Solve the problem. 1. On a sunny day, a flag pole and its shadow form the sides of a right triangle. If the hypotenuse is 0 meters long and the shadow is meters, how tall is the flag pole? m m 1 m m

3 . To measure the width of a river, a surveyor starts at point A on one bank and walks feet down the river to point B. He then measures the angle ABC to be '1''. Estimate the width of the river to the nearest foot. See the figure below. C. The angle of elevation from a point on the ground to the top of a tower is The angle of elevation from a point 1 feet farther back from the tower is 1. Find the height of the tower. Round to the nearest foot. 00 ft 19 ft 11 ft 00 ft Find the exact value without using a calculator.. csc - - A ft B ft 8 ft 0 ft 10 ft sec - -. An airplane travels at 180 km/h for hr in a direction of 89 from Greenville. At the end of this time, how far west of Greenville is the plane (to the nearest kilometer)? 10 km 9 km 9 km 81 km. A ship travels 99 km on a bearing of, and then travels on a bearing of 1 for 19 km. Find the distance from the starting point to the end of the trip, to the nearest kilometer. 81 km km 1 km 8 km. Find h as indicated in the figure. Round to the nearest foot Find the length of an arc intercepted by a central angle in a circle of radius r. Round your answer to 1 decimal place. 9. r = 1.9 ft; = 9 radians. ft 1. ft 0.9 ft. ft 0. r = 11.1 in.; = 1 8. in. 1. in..8 in. 10. in. 1. Find the distance between City E, N and City F, S. (Round to the nearest kilometer.) km 1,09 km 1,0 km km ft 8 ft 0 ft ft ft Assume that the cities lie on the same north-south line and that the radius of the earth is 00 km.. Find the latitude of Winnipeg, Canada if Winnipeg and Austin, TX, 0 N, are km apart. 0 N 0 N 0 N 0 N

4 . A wheel with a 8-inch radius is marked at two points on the rim. The distance between the marks along the wheel is found to be 1 inches. What is the angle (to the nearest tenth of a degree) between the radii to the two marks? Find the exact circular function value. 9. tan - -. Two wheels are rotating in such a way that the rotation of the smaller wheel causes the larger wheel to rotate. The radius of the smaller wheel is.1 centimeters and the radius of the larger wheel is 1. centimeters. Through how many degrees (to the nearest hundredth of a degree) will the larger wheel rotate if the smaller one rotates 10? Find the area of a sector of a circle having radius r and central angle. If necessary, express the answer to the nearest tenth.. r =.0 ft, = radians 0. csc The figure shows an angle in standard position with its terminal side intersecting the unit circle. Evaluate the indicated circular function value of. 1. Find sin. - 1, ft 0. ft.0 ft.1 ft. r = 19. mi, = mi. mi 01.1 mi.9 mi. Find the measure (in radians) of a central angle of a sector of area square inches in a circle of radius inches. Round to the nearest hundredth..8 radians 0.9 radians. radians 1.88 radians 8. A pendulum swings through an angle of 19 each secon If the pendulum is 1 cm in length and the complete swing from right to left lasts seconds, what area is covered by each complete swing? Round to the nearest hundredth..9 cm 191. cm 9.8 cm. cm

5 . Find csc. Suppose an arc of length s lies on the unit circle x + y = 1, starting at point (1, 0) and terminating at the point (x, y). Use a calculator to find the approximate coordinates (x, y). Round coordinates to four decimal places when appropriate., sec Use a table or a calculator to evaluate the function. Round to four decimal places.. csc s =. (0.99, 0.1) (-0.1, -0.99) (-0.1, 0.99) (0.1, 0.99) Find the exact values of s in the given interval that satisfy the given condition.. [0, ); tan s = 1,,,,,,,, 11. [-, ); cos s = 1,,, -, -,, -, -, -, - -, -,,

6 8. Let angle POQ be designated. Angles PQR and VRQ are right angles. If =, find the exact length of OQ. 0. Let angle POQ be designated. Angles PQR and VRQ are right angles. If =, find the length of OU accurate to four decimal places Let angle POQ be designated. Angles PQR and VRQ are right angles. If = 80, find the length of OQ accurate to four decimal places. 1. = radian per min, t = 1 min radians 1 radians radian 8 1 radians Use the formula = t to find the value of the missing variable. Give an exact answer unless otherwise indicate. = 9.0 radians per min, = 1.09 radians (Round to four decimal places when necessary.) 1.18 min 0.01 min.0 min 10.8 min Use the formula v = r to find the value of the missing variable. Give an exact answer unless otherwise indicate. v = 1 ft per sec, r =. ft (Round to four decimal places when necessary.) 0.9 radian per sec.099 radians per sec 0.0 radian per sec.88 radians per sec

7 Use the formula s = rt to find the value of the missing variable. Give an exact answer.. s = m, r = m, t = sec radian per sec radians per sec 1 radians per sec 1 radian per sec. A wheel is rotating at 8 radians/sec, and the wheel has a 8-inch diameter. To the nearest foot, what is the speed of a point on the rim in ft/min? ft/min ft/min 0 ft/min 0 ft/min. A wheel with a -inch diameter is turning at the rate of 8 revolutions per minute. To the nearest inch, what is the speed of a point on the rim in in./min? 0 in./min 009 in./min 0 in./min 01 in./min Graph the function.. y = sin 1 x

8 8. y = cos 1 x 8

9 9. y = + sin x + Graph the function over a one-period interval. 0. y = + sin(x - ) 9

10 . (sec + tan ) = 1 + sin 1 - sin MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Identities to find the exact value.. cos (- ) cos cos Find cos(s + t) given that cos s = 1, with s in quadrant I, and sin t = - 1, with t in quadrant IV SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 8. Find cos(s - t) given that cos s = -, with s in quadrant II, and cos t =, with t in 1 Verify that each equation is an identity. cos 1. sec + tan = 1 - sin. sec - 1 tan = tan sec + 1 quadrant IV

11 9. sin Find the exact value of the expression using the provided information.. Find sin(s - t) given that cos s = 1, with s in quadrant I, and sin t = - 1, with t in 0. tan sin 11 1 quadrant IV sin Find sin(s + t) given that cos s = -, with s in 1 quadrant II, and sin t = 1, with t in 1 quadrant II Use identities to find the indicated value for each angle measure.. tan Use a sum or difference identity to find the exact value.. sin 1. tan sin =, cos > 0 Find cos(). 9. sin = -, < < Find cos(). - Find the exact value by using a half-angle identity. 80. sin

12 81. cos tan sinx = sin x,,,,, 0,,, 8. sin Solve the equation (x in radians and in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. 89. sin x + sin x = 1 + n, + n 8. cos n, + n, + n + n, + n n, + n, + n 8. tan Give the exact value of the expression. 8. cos arcsin + arccos Solve the equation for exact solutions over the interval [0, ). 8. cosx + cos x + 1 = 0 } {},, 90. cos + cos = 1 { n, n} { n, n, n} { n, n, n, n} { n, n, n,. + 0 n} Solve the equation for solutions in the interval [0, ). 91. sin x = 1,,, 1,, 1 1,, 19 1 {0}, 0,, 1

13 9. cos x = - cos x,,, 8, 9 8, 8, sin-1x + tan-1x = 0 - -,, 0 1 0,,, Solve the equation for solutions in the interval [0, 0 ). Round to the nearest degree. 9. sin = cos {1, 1, 19, } {0, 90, 10, 0 } {0, 10, 180, 0 } {10, 1, 8, } Solve the equation for solutions over the interval [0, ). Write solutions as exact values or to four decimal places, as appropriate. 9. sin x + cos x = {0, } } 9. tan x + sec x = {0.,.9} {.1, 8.9} {1.10,.88} {0.18,.} Solve the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate. 99. B = 0.9 C = 11. b = 1.8 A =., a =., c = 1. A =., a = 1., c =. A =., a =., c = 11. A =., a = 11., c = A = 10' B = 10' a =. C = 11 0', b =., c =. C = 11 0', b =., c =. C = 11 0', b =., c =. C = 11 0', b =., c =. Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary A = 8. b = 1. in. c =. in.. in.. in. 1. in. 19. in. Solve the equation for exact solutions. 9. arcsin x + arccos x = arcsin x + arctan x = 0 - -, 1,,, 10. A = 0' b = 1. m c = 8.9 m.8 m.9 m 1 m 9.8 m Solve the problem. 10. Two tracking stations are on the equator 1 miles apart. A weather balloon is located on a bearing of N E from the western station and a bearing of N 1 E from the eastern station. How far, to the nearest mile, is the balloon from the western station? Round to the nearest mile. 1 mi 80 mi 8 mi mi 1

14 10. An airplane is sighted at the same time by two ground observers who are miles apart and both directly west of the airplane. They report the angles of elevation as 1 and 0. How high is the airplane? Round to the nearest hundredth of a mile. 1.9 mi 1. mi 0.8 mi 0. mi Find the missing parts of the triangle. 10. B = 19. b = 1.80 a = If necessary, round angles to the nearest tenth and side lengths to the nearest hundredth. A1 = 0.01, C1 = 10.9, c1 = 8.9; A = 19.99, C = 10.1, c =.8 A = 19.99, C = 10.1, c =.8 no such triangle A = 0.01, C = 10.9, c = C = 0' a = 18. c = 1.1 If necessary, round side lengths to the nearest hundredth. A1 = ', B1 = 10 0', b1 =.; A = 1 ', B = ', b =. A = ', B = 10 0', b =.19 no such triangle A1 = 10 0', B1 = ', b1 = 1.; A = ', B = 1 ', b =.19 Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate. 10. C = 10. a =. km b = 8.1 km c = 11. km, A = 1., B =. c = 1. km, A = 9., B =. c = 1. km, A =., B = 0. No triangle satisfies the given conditions C = 118. a =. m b = 11. m c =. m, A = 0.8, B = 0. c = 1. m, A =.8, B = 8. No triangle satisfies the given conditions. c = 19. m, A =.8, B = a = 18.9 cm b = 1. cm c = 1.9 cm 1 cm 11 cm 11 cm 10 cm 110. Two ships leave a harbor together traveling on courses that have an angle of 19 between them. If they each travel 0 miles, how far apart are they (to the nearest mile)? 181 mi mi 0 mi 90 mi 111. Two airplanes leave an airport at the same time, one going northwest (bearing 1 ) at 1 mph and the other going east at 8 mph. How far apart are the planes after hours (to the nearest mile)? 19 mi 9 mi 98 mi mi Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,0 ]. 11., ; ; ; ; 11. -, - 1 ; 1 1; 1; ; Two forces act at a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant force. 11. forces of 8.1 and. lb, forming an angle of. (round to the nearest pound) 089 lb lb lb 1 lb 1

15 Find the dot product for the pair of vectors , -1, -8, i - j, 8i + j - 0 Find the angle between the pair of vectors to the nearest tenth of a degree. 11.,, 9, i - j, i - 9j Determine whether the pair of vectors is orthogonal ,, -8, - Yes No 10. i - j, -8i - 1j Yes No 11. Two forces of 98 newtons and newtons act at a point. The resultant force is 8 newtons. Find the angle between the forces A force of 1 lb is required to pull a boat up a ramp inclined at 19 with the horizontal. How much does the boat weigh? 9 lb 8 lb 190 lb 0 lb Find the product. Write the product in rectangular form, using exact values. 1. [(cos 0 + i sin 0 )] [(cos 0 + i sin 0 )] 1 + 1i i i Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. (cos 00 + i sin 00 ) 1. (cos 0 + i sin 0 ) i - + i i i 8(cos 90 + i sin 90) (cos 0 + i sin 0) i 1 + i + i + i Find the given power. Write answer in rectangular form. 1. (- + i) i - i - + i i i 1 - i i i 1. Two boats are pulling a disabled vessel toward the landing dock with forces of 90 lb and 90 l The angle between the forces is 1.8. Find the direction and magnitude of the equilibrant. 18 lb at an angle of 10.8 with the 90-lb force 18 lb at an angle of 9. with the 90-lb force 18 lb at an angle of 19. with the 90-lb force 18 lb at an angle of 19. with the 90-lb force Find all specified roots. 19. Cube roots of 1. 1, 1 + i, i 1, i, i 1, 1 - i, i -1, 1 1

16 10. Cube roots of i. + 1 i, i, i - 1 i, i, i - 1 i, i, -i + 1 i, i, -i 1

17 Answer Key Testname: GENERAL TRIG REVIEW 1. d. b. d. c. a. b. c 8. c 9. b 10. a 11. d 1. b 1. c 1. d 1. d 1. b 1. c 18. c 19. d 0. b 1. d. a. d. c. c. d. d 8. b 9. b 0. a 1. b. c. d. a. c. c. d 8. c 9. d 0. b 1. b. c. d. c. d. d. d 8. d 9. a 0. a 1

18 Answer Key Testname: GENERAL TRIG REVIEW 1. b. a. d. a. c. b. b 8. b 9. a 0. b 1. sec + tan =. sec - 1 tan 1 sin 1 + sin + = cos cos cos = sec - 1 tan = 1 + sin cos sec + 1 sec + 1 = sec - 1 tan (sec + 1) = 1 - sin 1 - sin = 1 - sin cos (1 - sin ) = tan tan (sec + 1) =. (sec + tan ) = sec + sec tan + tan 1 sin = cos + cos. b. d. c. a 8. c 9. b 0. d 1. c. b. d. a. c. d. c 8. c 9. a 80. b 81. a 8. d 8. b 8. b 8. c 8. b 8. a 88. d 89. d 90. b 91. a 9. c 9. b (1 + sin ) 1 + sin = (1 - sin )(1 + sin ) 1 - sin 18 + sin cos tan sec + 1 cos cos (1 - sin ) = = 1 + sin + sin cos cos 1 - sin (1 + sin ) = 1 - sin =

19 Answer Key Testname: GENERAL TRIG REVIEW 9. d 9. d 9. c 9. d 98. b 99. d 100. c 101. d 10. b 10. c 10. b 10. a 10. a 10. a 108. b 109. c 110. d 111. b 11. a 11. b 11. c 11. b 11. a 11. c 118. b 119. b 10. a 11. a 1. c 1. d 1. c 1. a 1. c 1. d 18. d 19. b 10. d 19