MATH 135 Winter 2017: Assignment 9 Due at 8:25 a.m. on Wednesday, March 22, 2017 It is important that you read the assignment submission instructions and suggestions available on LEARN. 1. Shade the region of the complex plane defined by {2z 1 C : |z 3| 2}. Justify your answer. March 20 at 1pm: Written in a more standard way, the set above is {2z 1 : z C and |z 3| 2}. 2. Let z1 , z2 C and r R. Prove the following identity. |z1 |2 [(1 r) |z1 z2 |] + |z2 |2 (r |z1 z2 |) = |z1 z2 |(|(1 r)z1 + rz2 |2 + r (1 r) |z1 z2 |2 ). (When 0 r 1, this is Stewart's Theorem which you may have seen in the first or second lecture!) 3. Suppose n N and z C with |z| = 1 and z 2n 6= 1. Prove that zn 1+z 2n R. 4. Prove there exists m R such that the equation 2z 2 (3 3i)z (m 9i) = 0 has a real root. 5. Let z = 1 2 i 12 . Express z 26 in standard form. Show your work. 6. Use De Moivre's Theorem (DMT) to prove cos(5) = 16 cos5 20 cos3 + 5 cos for all R. You may look up and apply the Binomial Theorem to simplify an expression of the form (x + y)5 where x and y are complex numbers. \u0001 7. Let z be a nonzero complex number satisfying z +z 1 = 2 cos 15 . Determine the value of z 45 +z 45 . Give an exact answer. Show your work