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Trigonometry Review Test 2 Find the exact value of the expression. 3 1) cos-1 2 8 11 Find the inverse function f-1 of the function f. 15) f(x) = 5 sin x - 7 2) tan-1 1 Use a calculator to find the value of the expression rounded to two decimal places. 3) cos-1 (0.6) 4) sin-1 - 14) sin sin-1 16) f(x) = 7 cos x + 3 17) f(x) = 6 tan(8x) Find the domain of the function f and of its inverse function f-1 . 3 4 18) f(x) = 5 sin x - 7 5) tan-1 (-1.5) 19) f(x) = 3 tan x + 6 Find the exact value of the expression. Do not use a calculator. 5π 6) sin-1 sin 7 6π 7) cos-1 cos 7 4π 8) tan-1 tan 5 4π 9) cos-1 cos 3 π 10) cos-1 cos - 4 11) cos-1 cos - 7π 6 Find the exact value, if any, of the composite function. If there is no value, say it is ʺnot definedʺ. Do not use a calculator. 12) tan(tan-1 (-9.6)) 13) cos[cos-1 (-1.3)] 20) f(x) = 2 sin(5x) Find the exact solution of the equation. π 21) sin-1 x = 2 22) 3 sin-1 x = π 23) 4 cos-1 x = π 24) 3 tan-1 (2x) = π Find the exact value of the expression. 3 25) sec sin-1 - 2 26) cos sin-1 1 2 1 27) tan cos-1 - 2 2 28) sin cos-1 - 2 29) cot-1 -1 30) csc-1 -2 46) sin α + sin β = sin α sin β csc α + csc β 47) cot x csc x - 1 = 1 + csc x cot x 48) cot 2 x 1 - sin x = csc x + 1 sin x 31) cot-1 3 Use a calculator to find the value of the expression in radian measure rounded to two decimal places. 5 32) csc-1 2 Write the trigonometric expression as an algebraic expression in u. 33) sin (tan-1 u) Find the exact value of the expression. 11π 49) sin 12 34) cos (tan-1 u) 50) sin 15° 35) cos (sin-1 u) 51) sin 255° 36) cos (cot-1 u) Simplify the expression. cos θ 37) + tan θ 1 + sin θ Establish the identity. π 52) cos x + = -sin x 2 π 53) tan + x = -cot x 2 38) (1 + cot θ)(1 - cot θ) - csc2 θ 54) tan(θ - π) = tan θ Establish the identity. 39) tan u(csc u - sin u) = cos u 55) sin(α - β) = cot β - cot α sin α sin β 40) (sin x)(tan x cos x - cot x cos x) = 1 - 2 cos 2 x 56) cos(x - y) - cos(x + y) = 2 sin x sin y 41) (1 - cos x)(1 + cos x) = sin2 x 57) 42) tan u - 1 1 - cot u = tan u + 1 1 + cot u 43) sec θ - 1 tan θ = tan θ sec θ + 1 44) cot u + csc u - 1 = csc u + cot u cot u - csc u + 1 45) csc u - sin u = cos u cot u cos(x - y) 1 + tan x tan y = cos(x + y) 1 - tan x tan y Find the exact value of the expression. 3 1 58) sin cos-1 - sin-1 2 2 4 3 59) cos tan-1 - sin-1 3 5 2 1 60) sin sin-1 + cos-1 3 3 3 1 61) tan tan-1 + sin-1 4 2 Express the sum or difference as a product of sines and/or cosines. 75) sin(10θ) + sin(4θ) Write the trigonometric expression as an algebraic expression containing u and v. 62) cos(sin-1 u - cos-1 v) 76) cos(8θ) - cos(2θ) 77) cos(5θ) + cos(3θ) 63) sin(tan-1 u + tan-1 v) 78) sin(8θ) - sin(4θ) Establish the identity. 64) sec(2θ)= Establish the identity. sin(10θ) + sin(4θ) 79) = tan(7θ) cos(10θ) + cos(4θ) csc2 θ csc2 θ - 2 u csc u + cot u 65) cot2 = 2 csc u - cot u 80) α - β cos α + cos β = cot 2 sin α - sin β 66) cos(4u) = 2 cos2 (2u) - 1 Use the information given about the angle θ, 0 ≤ θ ≤ 2π, to find the exact value of the indicated trigonometric function. 5 3π θ 67) sin θ = - , < θ < 2π Find sin . 2 2 5 68) csc θ = - 6, cos θ > 0 θ Find cos . 2 3 69) cos θ = - , sin θ > 0 5 θ Find cos . 2 13 π 70) sec θ = - , < θ < π 12 2 θ Find sin . 2 Express the product as a sum containing only sines or cosines. 71) sin(9θ) cos(5θ) Solve the equation on the interval 0 ≤ θ < 2π. 81) 2 cos θ + 3 = 2 82) 4 sin2 θ = 1 3 θ 83) tan = 3 2 84) 2 3 sin(4θ) = 3 85) 7 csc θ - 1 = 6 Solve the equation. Give a general formula for all the solutions. 2 86) cos θ = - 2 Use a calculator to solve the equation on the interval 0 ≤ θ < 2π. Round the answer to two decimal places. 87) tan θ = 2.3 72) cos(3θ) cos(2θ) 88) 7 tan θ - 2 = 0 73) sin(5θ) sin(2θ) 74) cos(4θ) cos(7θ) Solve the equation on the interval 0 ≤ θ < 2π. 89) sin2 θ + sin θ = 0 90) cos2 θ - 1 = 0 91) sin2 θ = 5(cos θ + 1) 92) cos2 θ = 3(1 - sin θ) 93) cos2 θ - sin2 θ = 1 + sin θ 94) 1 + cos θ = 2 sin2 θ 95) tan(2θ) - tan θ = 0 96) sec θ = cos θ 105) A tree casts a shadow of 26 meters when the angle of elevation of the sun is 24°. Find the height of the tree to the nearest meter. Solve the triangle. 106) A = 60°, B = 100°, a = 1 107) B = 20°, C = 30°, a = 2 Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. 108) a = 7, b = 9, B = 49° 97) cos(2θ) - 3 cos θ + 2 = 0 109) b = 3, c = 5, B = 70° 98) cos θ - sin θ = 0 99) 3 sin θ - cos θ = -1 Solve the right triangle using the information given. Round answers to two decimal places, if necessary. 100) a = 8, B = 30°; Find b, c, and A. 101) a = 3, A = 20°; Find b, c, and B. 102) b = 5, c = 8; Find a, B, and A. Solve the problem. 103) A radio transmission tower is 150 feet tall. How long should a guy wire be if it is to be attached 5 feet from the top and is to make an angle of 20° with the ground? Give your answer to the nearest tenth of a foot. 104) A straight trail with a uniform inclination of 11° leads from a lodge at an elevation of 600 feet to a mountain lake at an elevation of 9000 feet. What is the length of the trail (to the nearest foot)? 110) a = 26, b = 20, B = 15° Solve the problem. 111) An airplane is sighted at the same time by two ground observers who are 3 miles apart and both directly west of the airplane. They report the angles of elevation as 11° and 23°. How high is the airplane? 112) A ship sailing parallel to shore sights a lighthouse at an angle of 10° from its direction of travel. After traveling 5 miles farther, the angle is 23°. At that time, how far is the ship from the lighthouse? 113) To find the distance AB across a river, a distance BC of 1034 m is laid off on one side of the river. It is found that B = 104.0° and C = 13.9°. Find AB. Round to the nearest meter. Solve the triangle. 114) b = 2, c = 3, A = 95° 115) a = 5, c = 10, B = 109° 116) b = 5, c = 6, A = 95° 117) a = 9, b = 9, c = 6 118) a = 8, b = 6, c = 4 Solve the problem. 119) Two points A and B are on opposite sides of a building. A surveyor selects a third point C to place a transit. Point C is 53 feet from point A and 70 feet from point B. The angle ACB is 51°. How far apart are points A and B? 120) A famous golfer tees off on a long, straight 455 yard par 4 and slices his drive 13° to the right of the line from tee to the hole. If the drive went 283 yards, how many yards will the golferʹs second shot have to be to reach the hole? Answer Key Testname: REVIEW TEST 2 1) π 6 2) π 4 27) - 3 2 28) 2 29) 3) 0.93 4) -0.85 5) -0.98 2π 6) 7 7) 30) - 31) 6π 7 8) - π 6 π 6 32) 0.41 u 33) u2 + 1 π 5 9) 2π 3 34) 10) π 4 35) 11) 3π 4 36) 5π 6 1 u2 + 1 1 - u2 u u2 + 1 37) sec θ 12) -9.6 13) not defined 8 14) 11 15) f-1 (x) = sin-1 x + 7 5 16) f-1 (x) = cos-1 x - 3 7 38) -2 cot2 θ 39) tan u(csc u - sin u) = tan u · csc u - tan u · sin u = 1 sin u 1 sin2 u sin u · - · sin u = - = cos u cos u cos u sin u cos u 1 - sin2 u cos2 u = = cos u cos u cos u 1 x 17) f-1 (x) = tan-1 8 6 40) (sin x)(tan x cos x + cot x cos x) = sin x sin x cos x cos 2 x - = sin 2 x - cos 2 x sin x cos x 18) Domain of f: (-∞, ∞) Domain of f-1 : [-12, -2] = (1 - cos 2 x)- cos 2 x = 1 - 2 cos 2 x. 41) (1 - cos x)(1 + cos x) = 1 - cos2 x = sin2 x 19) Domain of f: x ≠ (2k + 1)π ; k an integer 2 Domain of f-1 : (-∞, ∞) 20) Domain of f: (-∞, ∞) Domain of f-1 : [-2, 2] 21) x = 1 22) x = 3 2 23) x = 2 2 24) x = 3 2 25) 2 26) 3 2 1 - 1 cot u 1 - cot u cot u 1 - cot u 1 + cot u 42) tan u - 1 = tan u + 1 43) sec θ - 1 sec θ - 1 sec θ + 1 sec 2 θ - 1 = · = = tan θ tan θ sec θ + 1 tan θ(sec θ + 1) 1 + 1 cot u = 1 + cot u cot u tan θ tan2 θ = tan θ(sec θ + 1) sec θ + 1 = Answer Key Testname: REVIEW TEST 2 44) cot u + csc u - 1 cot u + (csc u - 1) = = cot u - csc u + 1 cot u - (csc u - 1) sin ((π/2) + x) π = 53) tan + x = cos ((π/2) + x) 2 cot u + (csc u - 1) cot u + (csc u - 1) · = cot u - (csc u - 1) cot u + (csc u - 1) 1 · cos x + sin x · 0 sin (π/2) cos x + sin x cos (π/2) = 0 · cos x - 1 · sin x cos (π/2) cos x - sin (π/2) sin x cot2 u + 2 cot u(csc u - 1) + (csc2 u - 2 csc u + 1) = cot2 u - (csc2 u - 2 csc u + 1) = -cot x. 54) tan (θ - π) = csc2 u - 1 + 2 cot u(csc u - 1) + (csc2 u - 2 csc u + 1) = csc2 u - 1 - (csc2 u - 2 csc u + 1) 55) 2csc2 u - 2 csc u + 2 cot u(csc u - 1) = -2 + 2 csc u 2 csc u(csc u - 1) + 2 cot u(csc u - 1) = 2 (csc u - 1) 2(csc u + cot u)(csc u - 1) = csc u + cot u 2 (csc u - 1) 45) csc u - sin u = cos u · 46) 48) 56) cos (x - y) - cos (x + y) = cos x cos y + sin x sin y - ( cos x cos y - sin x sin y) = 2 sin x sin y. cos (x - y) cos x cos y + sin x sin y 1/(cos x cos y) = = 57) cos (x + y) cos x cos y - sin x sin y 1/(cos x cos y) · cos x cos y + sin x sin y = cos x cos y - sin x sin y 1 + tan x tan y . 1 - tan x tan y cos u = cos u cot u sin u 58) 0 24 59) 25 60) 2 + 2 10 9 cot x cot x cot x csc 2 x - 1 = = = 1 + csc x (1+ csc x) cot x (1+ csc x) cot x 61) 9 + 4 3 12 - 3 3 (csc x + 1)(csc x - 1) csc x - 1 = . cot x (1+ csc x) cot x 62) v 1 - u2 + u 1 - v2 u + v 63) 2 u + 1 · v2 + 1 (sin α + sin β) · 47) sin α cos β - cos α sin β sin α cos β sin(α - β) = = sin α sin β sin α sin β sin α sin β cos α cos α sin β cos β = - = cot β - cot α sin β sin α sin α sin β 1 1 - sin2 u cos2 u - sin u = = = sin u sin u sin u sin α + sin β sin α + sin β sin α + sin β = = = 1 sin β + sin α 1 csc α + csc β + sin α sin β sin α sin β tan θ - tan π tan θ - 0 = = tan θ 1 + tan θ tan π 1 + tan θ · 0 sin α sin β = sin α sin β sin β + sin α cot 2 x csc2 x - 1 (csc x + 1)(csc x - 1) = = = csc x csc x + 1 csc x + 1 csc x + 1 sin x 1 - sin x 1 - = . 1 = sin x sin x sin x 49) 2( 3 - 1) 4 50) 2( 3 - 1) 4 51) - 2( 3 - 1) 4 52) cos x + π π π = cos x cos - sin x sin = (cos x)(0) 2 2 2 (sin x)(1) = - sin x. 64) sec(2θ) = 1 1 = = cos(2θ) 1 - 2 sin2 θ 1 sin2 θ 1 - 2 sin2 θ = csc2 θ csc2 θ - 2 u 65) cot2 = 2 1 u tan2 2 = 1 + cos u csc u + cot u = 1 - cos u csc u - cot u 66) cos(4u) = cos[2(2u)] = 2 cos2 (2u) - 1 67) 5 - 2 5 10 68) - 6 + 30 12 Answer Key Testname: REVIEW TEST 2 69) 5 5 93) 0, π, 7π 11π , 6 6 70) 5 26 26 π 5π 94) , π, 3 3 71) 1 [sin(14θ) + sin(4θ)] 2 72) 1 [ cos θ + cos(5θ)] 2 95) 0, π 96) 0, π π 5π 97) 0, , 3 3 73) 1 [ cos(3θ) - cos(7θ)] 2 98) 74) 1 [cos(3θ) + cos(11θ)] 2 99) 0, 75) 2 sin(7θ) cos(3θ) 76) -2 sin(5θ) sin(3θ) 77) 2 cos(4θ) cos θ 78) 2 sin(2θ) cos(6θ) 2 sin (7θ) cos (3θ) sin (7θ) sin (10θ) + sin (4θ) = = 79) cos (10θ) + cos (4θ) 2 cos (7θ) cos (3θ) cos (7θ) = tan (7θ) 80) cos α + cos β = sin α - sin β cot 2 cos α + β α - β cos 2 2 2 sin α - β α + β cos 2 2 α - β 2 81) 2π 4π , 3 3 82) π 5π 7π 11π , , , 6 6 6 6 83) π 3 84) π π 2π 7π 7π 13π 5π 19π , , , , , , , 12 6 3 12 6 12 3 12 85) π 2 86) θ θ = 3π 5π + 2kπ, θ = + 2kπ 4 4 87) 1.16, 4.30 88) 0.28, 3.42 3π 89) 0, π, 2 90) 0, π 91) π π 92) 2 cos α - β 2 sin α - β 2 = = π 5π , 4 4 4π 3 100) b = 4.62 c = 9.24 A = 60° 101) b = 8.24 c = 8.77 B = 70° 102) a = 6.24 B = 38.68° A = 51.32° 103) 424.0 ft 104) 44,023 ft 105) 12 m 106) C = 20°, b = 1.14, c = 0.39 107) A = 130°, b = 0.89, c = 1.31 108) one triangle A = 35.94°, C = 95.06°, c = 11.88 109) no triangle 110) two triangles A1 = 19.66°, C1 = 145.34°, c1 = 43.95 or 111) 112) 113) 114) 115) 116) 117) 118) 119) 120) A2 = 160.34°, C2 = 4.66°, c2 = 6.28 1.08 mi 3.86 mi 281 m a = 3.75, B = 32.1°, C = 52.9° b = 12.6, A = 22°, C = 49° a = 8.14, B = 37.7°, C = 47.3° A = 70.5°, B = 70.5°, C = 39° A = 104.5°, B = 46.6°, C = 28.9° 55.1 ft 190.2 yd