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Practice Test 3 Sections 5.5, 6.1, 6.2, 6.3, 6.4, and 6.5 Determine whether the given x-value is a solution of the equation. 3π 1) cos 2x = - 2, x = 4 A) Yes Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. 13) tan x = 3.2 B) No 14) cos x = -0.83 2) cos x + 1 = sin x, x = 5π 4 A) Yes Solve the triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If there isn't a triangle possible write not possible. If there are two triangles possible, write the answers under the heading triangle 1 and triangle 2. 15) B) No Find all solutions of the equation. Give answers in radians. 3) sin x = 0 65° 4) 9 cos x + 6 2 = 7 cos x+ 5 2 9 Find all solutions of the equation. 5) tan x sec x = -2 tan x 45° Solve the equation on the interval [0, 2π). 3 6) sin 4x = 2 7) cos 2x = 16) 2 2 7 6 8) 2 sin2 x = sin x 8 9) sin2 x - cos2 x = 0 17) A = 30°, a = 21, b = 42 10) sin2 x + sin x = 0 18) B = 50° C = 104° b = 19 11) -tan2 x sin x = -tan2 x 12) sin2 2x = 1 19) B = 104°, b = 5, a = 23 20) B = 17°, b = 13.8, a = 15.73 21) b = 2, c = 3, A = 85° 1 Find the area of the triangle having the given measurements. Round to the nearest square unit. 22) A = 38°, b = 14 meters, c = 12 meters 30) 4, -3π 4 5 23) a = 8 inches, b = 11 inches, c = 5 inches 24) C = 115°, a = 4 yards, b = 5 yards -5 5 25) a = 22 yards, b = 12 yards, c = 13 yards -5 Solve the problem. 26) To find the distance AB across a river, a distance BC of 160 m is laid off on one side of the river. It is found that B = 109.3° and C = 15.8°. Find AB. Round to the nearest meter. 31) -4, 5π 4 5 27) Two airplanes leave an airport at the same time, one going N 35° W at 417 mph and the other going due east at 329 mph. How far apart are the planes after 4 hours (to the nearest mile)? -5 28) The distance from home plate to dead center field in Sun Devil Stadium is 401 feet. A baseball diamond is a square with a distance from home plate to first base of 90 feet. How far is it from first base to dead center field? 5 -5 32) (-4, -225°) Use a polar coordinate system to plot the point with the given polar coordinates. 9π 29) 2, 4 5 5 -5 -5 5 5 -5 -5 2 Find another representation, (r, θ), for the point under the given conditions. π 33) 7, 4 46) (x - 13)2 + y2 = 169 47) y2 = 3x This is not multiple choice. Give an answer for a - d. Convert the polar equation to a rectangular equation. 48) r = 7 a) r > 0 and 2π < θ < 4π b) r < 0 and 0 < θ < 2π c) r > 0 and -2π < θ < 0 c) r < 0 and 2π < θ < 4π 49) θ = 5π 6 50) r = 5 csc θ Select the representation that does not change the location of the given point. 34) (8, 140°) A) (-8, 500)° B) (-8, 230)° C) (8, 320)° D) (8, 500)° 51) r cos θ = 9 52) r = 6 cos θ + 4 sin θ Plot the complex number. 53) -4i Polar coordinates of a point are given. Find the rectangular coordinates of the point. 35) (-7, 120°) 36) 9, i 6 4 3π 4 2 37) (4, -180°) -6 -4 -2 2 4 6 R 2 4 6 R -2 The rectangular coordinates of a point are given. Find polar coordinates of the point. Express θ in radians. 38) (2, -2 3) -4 -6 54) -3 39) (0, - 3) i 6 40) (-5, 0) 4 41) (-4 2, -4 2) 2 -6 Convert the rectangular equation to a polar equation that expresses r in terms of θ. 42) y = 9 -4 -2 -2 -4 -6 43) x2 + y2 = 25 44) x = 7 45) 8x - 7y + 10 = 0 3 55) -6 - i 61) -5i i 6 Write the complex number in polar form. Express the argument in radians. 62) - 2 3 - 2i 4 2 -6 -4 -2 2 4 63) 3 - 3i 6 R -2 Write the complex number in rectangular form. 64) 9(cos 120° + i sin 120°) -4 -6 65) -5(cos 56) 5 2 - 5 2i 3π 3π ) + i sin 4 4 i 66) 9(cos π + i sin π) 10 Find the product of the complex numbers. Leave answer in polar form. 67) z 1 = 5(cos 35° + i sin 35°) 5 -10 -5 5 10 R z 2 = 4(cos 8° + i sin 8°) -5 68) z 1 = 8 cos π π + i sin 6 6 z 2 = 3 cos π π + i sin 2 2 -10 57) -4 + 3i 10 i Find the quotient z1 z2 of the complex numbers. Leave answer in polar form. 69) z 1 = 14(cos 40° + i sin 40°) 5 z 2 = 2(cos 8° + i sin 8°) -10 -5 5 R -5 -10 70) z 1 = 3 cos 7π 7π + i sin 4 4 z2 = 6 cos 9π 9π + i sin 4 4 71) z 1 = 5(cos 200° + i sin 200°) Find the absolute value of the complex number. 58) z = -3 + 6i z 2 = 4(cos 50° + i sin 50°) Use DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form. 72) 4(cos 15° + i sin 15°) 4 59) z = -15 + 4i Write the complex number in polar form. Express the argument in degrees. 60) -15 + 20i 4 73) 2 2 (cos 7π 7π 5 ) + i sin 4 4 5 Answer Key Testname: PRACTICE TEST 3 1) B 2) B 3) nπ 29) 5 4) x = 3π 5π + 2nπ or x = + 2nπ 4 4 5) x = 2π 4π + 2nπ or x = + 2nπ or x = nπ 3 3 6) π π 2π 7π 7π 13π 5π 19π , , , , , , , 12 6 3 12 6 12 3 12 7) π 7π 9π 15π , , , 8 8 8 8 8) 0, π, 9) -5 π 5π , 6 6 -5 π 3π 5π 7π , , , 4 4 4 4 10) 0, π, 5 30) 5 3π 2 11) 0, π π 3π 5π 7π 12) , , , 4 4 4 4 13) 1.27, 4.41 14) 2.55, 3.73 15) B = 70°, a = 6.77, c = 8.68 16) A = 58°, B = 47°, C = 75° 17) B = 90°, C = 60°, c = 36.4 18) A = 26°, a = 10.9, c = 24.1 19) no triangle 20) A1 = 19°, C1 = 144°, c1 = 27.7; -5 5 -5 31) A2 = 161°, C2 = 2°, c2 = 1.6 21) a = 3.5, B = 35°, C = 60° 22) 52 square meters 23) 17 square inches 24) 9 square yards 25) 65 square yards 26) 53 meters 27) 2652 miles 28) 343.3 feet 5 -5 5 -5 6 Answer Key Testname: PRACTICE TEST 3 32) 53) i 5 6 4 2 -6 -5 -4 -2 5 2 4 6 R 2 4 6 R 2 4 6 R -2 -4 -6 54) -5 33) a) 7, i 9 5 7 π b) -7, π c) 7, - π 4 4 4 d) -7, 6 4 13 π 4 2 34) D 7 -7 3 35) , 2 2 36) -6 -4 -2 -2 -9 2 9 2 , 2 2 -4 37) (-4, 0) 5π 38) 4, 3 -6 55) i 6 39) (- 3, 90°) 40) (5, π) 41) (8, 225°) 9 42) r = sin θ 4 2 43) r = 5 -6 44) r = 7 cos θ 45) r = -10 (8 cos θ - 7 sin θ) -4 -2 -2 -4 -6 46) r = 26 cos θ 47) r = 3 cot x cscx 48) x2 + y2 = 49 49) y = - 3 x 3 50) y = 5 51) x = 9 52) (x-3)2 + (y-2)2 = 13 7 Answer Key Testname: PRACTICE TEST 3 56) 72) 128 + 128 3i 73) -128 + 128i i 10 5 -10 -5 5 10 R -5 -10 57) 10 i 5 -10 -5 5 R -5 -10 58) 3 5 59) 241 60) 25(cos 126.9° + i sin 126.9°) 61) 5(cos 270° + i sin 270°) 7π 7π 62) 4 cos + i sin 6 6 63) 3 2 cos 64) 65) 7π 7π + i sin 4 4 9 9 3 i + 2 2 5 2 -5 2 i + 2 2 66) -9 67) 20(cos 43° + i sin 43°) 2π 2π 68) 24 cos + i sin 3 3 69) 7(cos 32° + i sin 32°) 2 3π 3π 70) cos + i sin 2 2 2 71) 5 (cos 150° + i sin 150°) 4 8