Math 140 Study Guide. Find the exact values of the indicated trigonometric functions. Write fractions in lowest terms. 1)

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1 Math 40 Study Guide Find the exact values of the indicated trigonometric functions. Write fractions in lowest terms. ) 0 4) If csc q =, find cot q. A) C) B) 8 Find sin A and cos A. A) sin A = 3 ; cos A = 4 B) sin A = 4 ; cos A = 3 ) A fire is sighted due west of lookout A. At lookout B, 4. miles due south of A, the fire is also sighted. The angle at B is 30.3e. How far is the fire from B (to the nearest tenth of a mile)? A) 8.4 mi B).4 mi C).4 mi 9.4 mi ) C) sin A = 4 ; cos A = 3 sin A = 4 3 ; cos A = Find tan A and cot A. A) tan A = ; cot A = B) tan A = ; cot A = C) tan A = ; cot A = 3 3 tan A = 3 3 ; cot A = Express the angle in degrees to the nearest hundredth. ) e' A).0 B). C).. Convert the angle measures to degrees, minutes, and seconds. Round seconds to whole units. ) 43.39e A) 43e3'30'' B) 43e3''' C) 43e3'4'' 43e3'39'' Write in terms of the cofunction. 8) sin 4e A) tan 49e B) cot 49e C) cos 4e cos 49e 9) cot 3e A) tan 49e B) cot 9e C) tan 3e tan 9e Find the requested function value of q. 3) If sin q = 8, find sec q. A) C) 8 B) 8

2 Solve the right triangle for all missing sides and angles to the nearest tenth. 0) Find the trigonometric function value of angle A. 4) Given that the terminal side of A passes through (, ), find sin A. A) B) C) Find the trigonometric function value for the angle shown. c = 3 B = 3e A) A = 8e, a = 9., b =. B) A = 8e, a =., b = 9. C) A = 8e, a = 9., b = 4.4 A = 8e, a = 4.4, b =. ) sec q Solve the right triangle. A) sec q = 8 8 B) sec q = 8 C) sec q = sec q = ) a = 8., b =.8 A) A = 38., B =.4, c = 3. B) A = 38., B =.4, c = 9. C) A =.4, B = 38., c = 9. A = 3, B = 3, c = 3. ) A blimp is 00 meters high in the air and measures the angles of depression to two stadiums to the west of the blimp. If those measurements are.e and.9e, how far apart are the two stadiums to the nearest meter? A) 39 m B) 30 m C) 44m 3 m 3) What is the angle of elevation of the sun when a 0-ft flag pole casts a -ft shadow? Round to the nearest tenth of a degree. A) 9.3e B) 0.e C) 9.e 0.e Find the trigonometric function value of angle A. ) cos A = and A in quadrant IV Find sin A. A) - B) - C) ) csc A = - Find cot A. A) - 4 C) - 4 and A in quadrant III B)

3 Find the measures of two angles, one positive and one negative, that are coterminal with the given angle. 8) 3e A) e; -44e B) e; -4e C) 39e; -34e 39e; -44e 9) Find the complement of an angle whose measure is 4.e. A).48e B).48e C) 4.e 4.e 0) Find the supplement of an angle whose measure is.3e. A) 0.3e B).3e C) 4.8e 4.8e Find the supplement or complement. ) Supplement of p A) p 4 C) p B) p p Convert to degree measure. Round to two decimal places, if necessary. 3 ) 9 p A) 30e B).e C) 80pe 40e ) A) 4.0e B) 4.0e C).0e.30e ) A bicycle wheel rotates times in minute. Through how many degrees does a point on the tip of the wheel move in seconds? A) 4e B) 3e C) 30e 4e 8) In a circle with a -ft radius, how long is an arc associated with an angle of 0. radians? A) 4. ft B). ft C). ft 30 ft Find the amplitude, period or phase shift. 9) Find the amplitude of y = cos (4x + p ). ) Complement of p 8 A) p 8 B) p C) 3p 8 p 8 A) B) 8 C) 4 p Convert to radian measure. Leave your answer in terms of p. 3) -88.3e A) 8.. p B) p C). 9. p p Convert to radian measure. Round to two decimal places. 4) -0.3e A) -3. B) -3.3 C) ) Find the period of y = 3 cos (3x + p ). A) p 3 B) p C) 3 p 3) Find the phase shift of y = - + sin(x - p ). A) p 4 to the right B) p to the left C) p to the left p to the right Multiply and simplify. 3) (cos x - sin x) A) cosx + sinx B) C) - sin x cos x cosx + sin x - sinx 3

4 Factor and simplify. 33) - sinx + sin4x A) sinx B) ( + tanx) C) ( - sinx) cos4x Find the exact value. 39) Given that sin A = -4/ with A in quadrant IV, find sin A. A) B) - 34) sinx - sin x + C) 4-4 A) sin x B) sin x - C) cosx sin x + Simplify the expression. 3) 4 cos3 x sin x sin x cos x A) 4 cos x tan x B) 4 cos x cot x C) 4 sin x cot x cos x cot x 4 40) Find an equivalent expression for cos 3x in terms of powers of cos x. A) cos3 x + 3 cos x B) 4 cos x - 3 cos x C) 9 cos x 4 cos3 x - 3 cos x Simplify. Check your result using a grapher. Evaluate exactly. 3) cos p A) ( 3 - ) B) C) - ( 3 - ) - ( 3 - ) 4 ( 3 - ) 4 4) - sin x A) sin x B) sin x C) cos x cos x Evaluate. 4) csc (sin- 3 ) 3) If cos q = 3 and cos f = 4, find cos (q + f). A) 3 B) C) 33 38) Given that sin q = 3 and that the terminal side A) 4 3 B) 3 C) ) sin (arctan ) A) B) / C) / Find. 3 is in quadrant II, find cot q and csc q. A) cot q = and csc q = ) cos arctan a A) a + 4 B) a + 4 B) cot q = and csc q = 3 C) a a + 4 a - 4 C) cot q = and csc q = - 4 cot q = and csc q = 3 4

5 Solve, finding all solutions. 3 4) sin x = (Express your answer in radians.) A) p + kp, where k is any integer 3 B) p 3 + kp and p 3 integer C) p + kp and p integer + kp, where k is any + kp, where k is any p + kp and - p + kp, where k is any integer Solve, finding all solutions in [0, p). 4) sin x = - sinx A) No solution B) x = p, p, p C) x = p, 3p, p x = p, p, 3p 4) A generator produces an alternating current according to the equation I = 48 sin 3pt, where t is time in seconds and I is the current in amperes. What is the smallest time t such that I = 4? A) 40 sec B) 80 sec C) 40 sec 0 sec Solve the triangle, if possible. 48) B = 0.9e C = 0.9e b = 8.9 A) A = 49.e, a = 0.8, c = 4.4 B) A =.e, a = 43.4, c =.8 C) A = 49.e, a =.8, c = 43.4 A =.e, a = 4.4, c = ) B = 8.e b = 4.38 a =. A) No solution B) A = 30e, C = 3.e, c = C) A = 0e, C =.e, c = 9.04 A = 30e, C = 3.e, c = 33.94; A' = 0e, C' =.e, c' = 9.04 Find the area of triangle ABC. 0) A = 34.e b = 3. in. c =.3 in. A) 0 in B) 3 in C) 8 in 9 in ) To find the distance AB across a river, a distance BC of 49 m is laid off on one side of the river. It is found that B = 0.e and C =.e. Find AB. A) m B) 04 m C) m 0 m Solve the triangle, if possible. ) a =. b = 3.3 c =. A) A = 9.84e, B = 9.0e, C = 9.4e B) A =.84e, B =.0e, C = 9.4e C) A =.84e, B =.0e, C = 93.4e No solution 3) C = 3.e a =.80 b = 8.0 A) c = 8.4, A =.e, B = 38.8e B) c =., A = 9.e, B = 3.8e C) c =., A = 3.e, B = 34.8e No solution

6 4) Two points, A and B, are on opposite sides of a building. A surveyor chooses a third point, C, yd from B and 00 yd from A, with angle ACB measuring 4.e. How far apart are A and B (to the nearest yard)? A) 0 yd B) yd C) 93 yd 0 yd ) Three ships, A, B, and C, are anchored in the ocean. The distance from A to B is. km, from B to C is. km, and from C to A is 3.4 km. Find the angle measurements of the triangle formed by the three ships. A) A = 8.9e; B = 3.e; C = 0.9e B) A = 3.e; B =.e; C = 3.e C) A = 8.9e; B = 9.9e, C = 3.e A = 3.e; B =.e; C = 9.e Find the absolute value of the complex number. ) + i A) 3.3 B).8 C) 4.9 Express the complex number in trigonometric form. ) 4-4i Express in standard notation. 9) 8(cos 30e + i sin 30e) A) i B) i C) i i Find standard notation a + bi. 0) 3 cos p + i sin p A) 3 + i B) i C) i + 3 i Multiply or Divide and leave the answer in trigonometric notation. ) cos p 3 + i sin p 3 œ cos p + i sin p A) cos p + i sin p B) 0 cos p + i sin p C) 0 cos p + i sin p cos p + i sin p A) 4 cos p 4 + i sin p 4 B) 4 cos p 4 + i sin p 4 C) 4 cos p 4 + i sin p 4 4 cos p 4 + i sin p 4 8) i A) 0 cos 3p + i sin 3p B) 3 cos 4p 3 + i sin 4p 3 C) 3 cos 3p + i sin 3p ) cos p 4 + i sin p 4 cos p + i sin p A) B) C) 4 cos 3p cos cos - 3p cos p + i sin 3p + i sin + i sin - 3p + i sin p 0 cos 4p 3 + i sin 4p 3

7 Convert to trigonometric notation and perform the indicated operation. + i 3) 3 + i A) cos e + i sin e B) ( - )(cos e + i sin e) C) (cos e + i sin e) (cos e + i sin e) Raise the number to the indicated power and express in trigonometric notation. 4) ( + i) A) 0(cos 4.00e - i sin 4.00e) B) 0(cos 4.00e + i sin 4.00e) C) 0(cos 90.00e + i sin 90.00e) 0(cos 90.00e - i sin 90.00e) ) 3 cos 4 p + i sin 4 p 4 A) 8 4cos 4 p + i 4sin 4 p 9) Square roots of - + 3i A) - i, - - i B) / + i/, - / - i/ C) + i, - - i / - i/, - / + i/ Find all solutions. 0) z - i = 0 A) / + i/, - / - i/ B) -i, i C) -, / - i/, - / + i/ Prove the identity. sin q ) + cos q + + cos q = csc q sin q ) tan x + cot x = csc x Determine whether the equation is an identity. If it is an identity, prove it. 3) cos x csc x tan x = B) 3 cos p + i sin p C) 3 cos 4 p + i sin 4 p 8 cos p + i sin p Raise the number to the given power and write standard notation for the result. ) - + i 3 A) i B) + i C) + i i ) (4(cos 0e + i sin 0e)) A) i B) i C) i i Find the indicated roots. 8) Cube roots of 8i A) i, 3 - i, i B) -i, 3 + i, i C) i, 3 + i, i -i, i, 3 - i

8 Answer Key Testname: MATH40STUDYGUIDE ) B ) B 3) B 4) D ) B ) A ) C 8) D 9) D 0) A ) B ) D 3) B 4) B ) B ) D ) B 8) C 9) A 0) C ) D ) C 3) B 4) D ) D ) D ) D 8) B 9) A 30) A 3) D 3) C 33) D 34) B 3) B 3) B 3) C 38) B 39) D 40) D 4) D 4) B 43) C 44) B 4) B 4) D 4) B 48) D 49) D 0) A 8 ) A ) B 3) B 4) C ) B ) B ) B 8) A 9) C 0) B ) B ) A 3) C 4) C ) D ) D ) D 8) B 9) B 0) A sin q ) + cos q + + cos q = sin q + ( + cos q) sin q ( + cos q)(sin q) = sin q + + cos q + cos q ( + cos q)(sin q) q. = ( + cos q) ( + cos q)(sin q) = csc sin x ) cos x + cos x sin x = sin x + cos x = sin x cos x sin x cos x = sin x cos x = = csc x. sin x 3) The statement is an identity. The student's answer should include a proof.