Question

Find the following product, and write the product in rectangular form, using exact values.

[8( cos 45° + i sin 45°)][7( cos 165° + i sin 165°)]
[8( cos 45° + i sin 45°)][7(cos 165° + i sin 165°)] = IN
(Type your answer in the form a + bi.)

Answer 1

Writing each complex number in exponential form makes this very easy. Recall Euler's formula:

`e^(i\theta)=\cos\theta+i\sin\theta`

Then

`8(\cos45^\circ+i\sin45^\circ)=8e^(i\pi/4)`

(since 45º = π/4 rad)

`7(\cos165^\circ+i\sin165^\circ)=7e^(i(11\pi/12))`

(since 165º = 11π/12 rad)

The product is

`56e^(i(\pi/4+11\pi/12))=56e^(i(7\pi/6))`

and in Cartesian form this is

`56(\cos210^\circ+i\sin210^\circ)=56\left(-\frac{\sqrt3}2-i\frac12\right)=\boxed{-28\sqrt3-28i}`