cos(θ) = = sin(θ) = = Therefore (x, is located in the quadrant II, and = . So . To put our answer in form: y) θ = = cos() 0.78824 a rcos(θ) ≈ = = sin() 1.903 b rsin(θ) ≈ Thus 0.78824 + 1.903i. 63. = . Let’s convert to polar form: 5 + 3i = Then = 5 and = 3. Then = + = x + yi. x y r2 x2 y2 52 + 32 = 34 so = (since 0). Also: r r ≥ = and = x rcos(θ) y rsin(θ) 5 = cos(θ) 3 = sin(θ) cos(θ) = sin(θ) = Therefore (x, is located in the quadrant I, and 0.54042. So . To put our answer in form: y) θ ≈ = = cos(0.18014) 1.771 a rcos(θ) ≈ = = sin(0.18014) 0.3225 b rsin(θ) ≈ Thus 1.771 + 0.3225i. 65. If = 2 then = . In the complex plane, 2 would sit on the horizontal axis at an angle of 0, z5 z giving the polar form . Then = e0 = cos (0) + sin(0) = 1.149. i ≈ Since the angles 2π, 4π, 6π, 8π, and 10π are coterminal with the angle of 0, can be represented by turns as , , , , and to get all solutions. = ( = = cos() + 0.355 + 1.092i isin() ≈ = ( = = cos() + -0.929 + 0.675i isin() ≈ = ( = = cos() + -0.929 - 0.675i isin() ≈ = ( = = cos() + 0.355 - 1.092i isin() ≈
cos(θ) = = sin(θ) = = Therefore (x, y) is located in the quadrant II, and θ = . So . To put our answer in form: a = rcos(θ) = cos() ≈ 0.78824 b = rsin(θ) = sin() ≈ 1.903 Thus 0.78824 + 1.903i. 63. = . Let’s convert to polar form: 5 + 3i = x + yi. Then x = 5 and y = 3. Then r2 = x2 + y2 = 52 + 32 = 34 so r = (since r ≥ 0). Also: x = rcos(θ) and y = rsin(θ) 5 = cos(θ) 3 = sin(θ) cos(θ) = sin(θ) = Therefore (x, y) is located in the quadrant I, and θ ≈ 0.54042. So . To put our answer in form: a = rcos(θ) = cos(0.18014) ≈ 1.771 b = rsin(θ) = sin(0.18014) ≈ 0.3225 Thus 1.771 + 0.3225i. 65. If z5 = 2 then z = . In the complex plane, 2 would sit on the horizontal axis at an angle of 0, giving the polar form . Then = e0 = cos (0) + i sin(0) = ≈ 1.149. Since the angles 2π, 4π, 6π, 8π, and 10π are coterminal with the angle of 0, can be represented by turns as , , , , and to get all solutions. = ( = = cos() + isin() ≈ 0.355 + 1.092i = ( = = cos() + isin() ≈ -0.929 + 0.675i = ( = = cos() + isin() ≈ -0.929 - 0.675i = ( = = cos() + isin() ≈ 0.355 - 1.092i
Chapter%208%20Solutions.docx