Question

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

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

$56(\cos210^\circ+i\sin210^\circ)=56\left(-\dfrac{\sqrt3}2-i\dfrac12\right)=\boxed{-28\sqrt3-28i}$