1. Let sin +i cos 3 3 (a) Express in polar form. (b) Hence, or otherwise,...

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1. Let sin +i cos 3 3 (a) Express in polar form. (b) Hence, or otherwise, find the three cubic roots of C, expressing your answer in polar form. 2. In this question, there is no need to justify your answer. = (a) Give a full and explicit description of all solutions of the equation 25-325-0 with complex unknown 2. Express your answer in polar form. (b) Write down all the complex solutions of the system of inequalities 25-32i = 0 Re(z) Im(s) () + i sin 3 (1) 3. Let f(z) be the polynomial given by f(z) = 210+25 +1. Let w = cos == (a) Write down the quintic roots of w. (b) Express the polynomial (2) first in the form (25+ P)(" + Q), and hence further in the form ( + Az + 1)( + B + 1)( + C + 1)(z + Dz + 1)(= + Ez + 1) in which P,Q are some appropriate complex numbers, and A, B, C, D, E are some appropriate real numbers. You have to give the values of P, Q, A, B, C, D, E explicitly. (c) By applying the result above, or otherwise, determine the values of the numbers below. Justify your answers. i. cos 15 (3) 8 4 26 + cos + cos + cos + cos 15 15 3 15 ii. + COS A-A-A-A-A-A-A-A- ()(1) () () () () () () () () +cos ( 2T 8 COS 15 15 1. Let sin +i cos 3 3 (a) Express in polar form. (b) Hence, or otherwise, find the three cubic roots of C, expressing your answer in polar form. 2. In this question, there is no need to justify your answer. = (a) Give a full and explicit description of all solutions of the equation 25-325-0 with complex unknown 2. Express your answer in polar form. (b) Write down all the complex solutions of the system of inequalities 25-32i = 0 Re(z) Im(s) () + i sin 3 (1) 3. Let f(z) be the polynomial given by f(z) = 210+25 +1. Let w = cos == (a) Write down the quintic roots of w. (b) Express the polynomial (2) first in the form (25+ P)(" + Q), and hence further in the form ( + Az + 1)( + B + 1)( + C + 1)(z + Dz + 1)(= + Ez + 1) in which P,Q are some appropriate complex numbers, and A, B, C, D, E are some appropriate real numbers. You have to give the values of P, Q, A, B, C, D, E explicitly. (c) By applying the result above, or otherwise, determine the values of the numbers below. Justify your answers. i. cos 15 (3) 8 4 26 + cos + cos + cos + cos 15 15 3 15 ii. + COS A-A-A-A-A-A-A-A- ()(1) () () () () () () () () +cos ( 2T 8 COS 15 15