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 z n 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 √ 1 2. Express z 26 in standard form. Show your work.

6. Use De Moivre’s Theorem (DMT) to prove cos(5θ) = 16 cos5 θ − 20 cos3 θ + 5 cos θ for all θ ∈ 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.

7. Let z be a nonzero complex number satisfying z+z −1 = 2 cos π 15
.

 Determine the value of z 45+z−45.