Originally Posted by

**andrew2322** Hello all, I have done a few questions that I need corrected.

1.) Sketch the region of the complex plane defined by Im (z^2) for pi/8 < arg (z) < 3pi/8

my solution: I let z = x + iy, then found z^2 and the imaginary parts of z^2 which gave me a graph shape of 1/x. the region defined by the angle is in the first quadrant and only a small bit of that curve is included.

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2.) z = 1/4 - [root(3) / 4]i

a) Write z in exponential polar form for angles between -pi and pi

my solution: 1/2 * e ^ i (pi/3)

b) Calculate (1/z) ^ 9

my solution: -2 * 2^8 + 0i

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3.) Use the complex exponential to express sin^3(theta) as a sum of sines or cosines of multiples of theta.

My solution: -1/i sin(3 theta) + 3/i cos (theta)

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Thank you in advance.