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FINAL EXAM REVIEW TRIGONOMETRY Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression without using a calculator or table. 3 1) cos-1 2 A) 7π 4 B) π 6 C) 2) arcsec ( 2) 7π A) 4 C) 1) π 4 D) 11π 6 2) 3π 4 B) π 7π ± 2πn, ± 2πn 4 4 D) π 4 Find the exact value of the expression in degrees without using a calculator or table. 3) csc-1 (2) B) -30° A) 60° C) 420° Find an equivalent algebraic expression for the composition. 4) cot (arctan (x)) 1 B) A) x x Find all real numbers that satisfy the equation. 5) cos x = 1 π A) x|x = + 2kπ 2 C) x|x = 6) cos 2x = 3) D) 30° 4) 1 C) x2 D) x 5) B) {x|x = π + 2kπ} 3π + 2kπ 2 D) {x|x = 0 + 2kπ} 3 2 6) π 5π A) x x = + kπ or x = + kπ 6 6 C) x x = π 11π + 2kπ or x = + 2kπ 12 12 B) x x = π + kπ 12 D) x x = π 11π + kπ or x = + kπ 12 12 Find all values of θ in [0°, 360°) that satisfy the equation. 1 7) sin θ = - 2 A) {60°, 300°} 7) B) {150°, 210°} C) {210°, 330°} 1 D) {60°, 120°} Find all real numbers in [0, 2π] that satisfy the equation. 3 8) sin 4x = 2 A) 0, C) π , π 4 B) π 5π , 4 4 9) cos 2x = A) 8) D) 0 2 2 9) π 3π 5π 7π , , , 4 4 4 4 C) 0, π π 2π 7π 7π 13π 5π 19π , , , , , , , 12 6 3 12 6 12 3 12 B) 2π 4π , π, 3 3 π 7π 9π 15π , , , 8 8 8 8 D) no solution Solve the triangle with the given parts. 10) 10) 92.4 B) β = 36.8°, b = 285, c = 355.7 D) β = 37.2°, b = 287.6, c = 355.7 A) β = 37.2°, b = 355.7, c = 287.6 C) β = 37.2°, b = 29.7, c = 24.1 Solve the triangle. If there is more than one triangle with the given parts, give both solutions. 11) β = 65.7° b = 4.42 a = 17.5 A) α = 33.85°, γ = 81.45°, c = 25.92 B) No solution D) α = 31.85°, γ = 81.45°, c = 23.92 C) α = 32.85°, γ = 81.45°, c = 21.92 11) 12) 12) α = 30.0° a = 12.79 b = 25.58 A) No solution C) β = 90.0°, γ = 60.0°, c = 22.1 B) β = 60.0°, γ = 60.0°, c = 22.1 D) β = 60.0°, γ = 90.0°, c = 22.1 Solve the triangle with the given information. 13) γ = 123.5° a = 4.80 b = 9.41 A) c = 18.5, α = 16.4°, β = 40.1° C) c = 15.6, α = 20.4°, β = 36.1° 13) B) c = 12.7, α = 18.4°, β = 38.1° D) No solution 2 14) 14) a = 8.9 b = 13.2 c = 16.1 A) α = 31.5°, β = 55.0°, γ = 93.4° C) α = 33.5°, β = 55.0°, γ = 91.4° B) α = 35.5°, β = 53.0°, γ = 91.4° D) No solution Solve the problem. 15) Find the area of the region. A) 25.02 ft 2 15) B) 6.26 ft 2 C) 12.51 ft 2 Find the area of the triangle using Heronʹs formula. Round to the nearest unit. 16) a = 64.6 b = 65.4 c = 67.2 A) 1869 B) 1947 C) 1961 D) 14.69 ft 2 16) D) 1941 Find the area of the triangle. If necessary, round the answer to two decimal places. 17) 17) 105° 2 A) 2.33 9 B) 8.69 C) 101.42 D) 34.77 The vectors u, v, and w are drawn below. Sketch the specified vector. 18) u + v A) 18) B) C) 3 D) 19) v + w A) B) 19) C) D) Find the magnitudes of the horizontal and vertical components for the vector v with the given magnitude and given direction angle. Round to an appropriate number of significant digits. 20) 20) ∣v∣ = 178.6, θ = 112° B) vx = -66.9, vy = 165.6 A) vx = 165.6, vy = 66.9 D) vx = 66.9, vy = 165.6 C) vx = 165.6, vy = -66.9 Find the magnitude and direction angle (to the nearest tenth) of the vector. Give the measure of the direction angle as an angle in [0°, 360°). 21) 21) 9, 9 A) 9; 225° B) 18; 45° C) 9 2; 45° D) 9 2; 225° 22) 21, -1 A) 22; 347.7° 23) 0, -18 A) 324; 270° 22) B) 22; 102.3° C) 22; 347.7° B) 18; 90° C) 18; 180° D) 22; 282.3° 23) D) 18; 270° Find the component form for the vector v with the given magnitude and direction angle θ. 24) v = 78.9, θ = 79.3° B) 14.6, 77.5 C) 77.5, 14.6 D) -14.6, -77.5 A) -77.5, -14.6 25) v = 153.8, θ = 121.6° A) 131.0, 80.6 25) C) -80.6, 131.0 B) 80.6, 131.0 Perform the indicated operation. Use the form <a, b> for vectors. 26) u = <9, 2>, v = <-12, -3>; Find u - v. A) <11, -15> B) <21, 5> C) <-3, -1> 27) v = <2, 10>; Find -1v. A) <-2, 10> 24) D) 131.0, -80.6 26) D) <7, -9> 27) B) <1, 10> 28) v = <2, 7>, u = <2, 5>; Find 8v + 9u. A) <648, 504> B) <32, 108> C) <1, 9> D) <-2, -10> C) <72, 63> D) <34, 101> 28) 4 Identify the complex number as real or imaginary. 29) 5i A) Real 29) B) Imaginary 30) 2π A) Real B) Imaginary 30) 31) -8 - 7i A) Real B) Imaginary 31) Express in terms of i. 32) -25 A) 5i 32) B) -5i C) -i 5 D) ±5 Perform the indicated operation and write the answer in the form a + bi, where a and b are real numbers. 33) (4 - 7i) + (7 + 3i) A) 11 + 4i B) 11 - 4i C) -11 + 4i D) -3 + 10i 34) 5i + (-7 - i) A) -7 + 6i B) 7 - 4i C) 7 - 6i D) -7 + 4i 35) (3 + 5i) - (-5 + i) A) -8 - 4i B) 8 + 4i C) -2 + 6i D) 8 - 4i 34) 35) 36) 9i(5 - 6i) A) 54 + 45i B) 45i - 54 37) (9 - 4i)(2 + 6i) A) 42 - 46i 38) (6 - 4i)2 A) 36 + 16i 36) C) 45i + 54i2 D) 45i - 54i2 B) 42 + 46i C) -6 - 62i D) -24i2 + 46i + 18 B) 52 - 48i C) 20 - 48i D) 36 - 16i A) -1 B) i C) -i D) 1 40) i42 A) -i B) 1 C) i D) -1 37) 38) Evaluate the indicated power of i. 39) i8 39) 40) Write the quotient in the form a + bi. 9 + 3i 41) 3 - 4i A) 39 27 + i 5 5 33) 41) 3 9 B) - + i 7 7 C) 5 3 9 + i 5 5 D) - 39 9 + i 7 7 42) 9 i 42) A) -9 43) B) -9i C) 9 D) 9i 4 7i 43) A) - 4 7 B) 4 7 C) 4 i 7 Write the expression in the form a + bi, where a and b are real numbers. 44) 10 + -81 A) 1 B) 10 - 9i C) 10 + 9i 45) 4 D) - i 7 44) D) 10 + i -16 -4 45) A) 2 C) -2i B) 4i D) 2i Find the absolute value of the complex number. Round your answer to two decimal places, if necessary. 46) 6 + 3i A) 6.71 B) 3.00 C) 4.18 D) 9 47) 3 - 3i A) 9 47) B) 4.24 C) 3.46 D) 2.45 Write the complex number in trigonometric form, using degree measure for the argument. 48) -5 A) 5(cos 270° + i sin 270°) B) 5(cos 0° + i sin 0°) C) 5(cos 180° + i sin 180°) D) 5(cos 90° + i sin 90°) 49) -15 + 20i A) 25(cos 126.9° + i sin 126.9°) C) 25(cos 53.1° + i sin 53.1°) 48) 49) B) 25(cos 306.9° + i sin 306.9°) D) 25(cos 233.1° + i sin 233.1°) Write the complex number in the form a + bi. 50) 8(cos 30° + i sin 30°) 1 3 B) + i A) 4 + 4 3i 4 4 51) 50) C) 3 1 + i 4 4 D) 4 3 + 4i 5 (cos 150° + i sin 150°) 2 A) - 46) 3 1 + i 4 4 51) B) - 3 1 + i 4 2 C) - 5 3 5 + i 4 4 Perform the indicated operation. Write the answer in the form a + bi. 52) 3(cos 30° + i sin 30°) · 2(cos 90° + i sin 90°) B) -3 + 3 3i C) -6 - 6 3i A) 2 6 - 2i 6 D) - 5 3 5 + i 4 2 52) D) 3 - 12 3i 53) 5(cos 200° + i sin 200°) 4(cos 50° + i sin 50°) A) -2 + 2 3i 53) 1 3 B) - + i 2 2 C) -10 + 10 3i 7 D) - 5 3 5 + i 8 8 Answer Key Testname: FINALEXAM_REVIEW 1) B 2) D 3) D 4) B 5) D 6) D 7) C 8) B 9) B 10) D 11) B 12) C 13) B 14) C 15) C 16) A 17) B 18) C 19) C 20) B 21) C 22) C 23) D 24) B 25) C 26) B 27) D 28) D 29) B 30) A 31) B 32) A 33) B 34) D 35) B 36) A 37) B 38) C 39) D 40) D 41) C 42) B 43) D 44) C 45) A 46) A 47) B 48) C 49) A 50) D 8 Answer Key Testname: FINALEXAM_REVIEW 51) C 52) B 53) D 9