Question

Find the quotient and write it in rectangular form using exact values: 8 ( cos pi/2 + i sin pi/2 ) /3 ( cos pi/6 + i sin pi/6 )

Answer 1

Answer:

$\frac{4}{3}+\frac{4\sqrt{3}}{3}i$

Explanation:

Given:

$\frac{8(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2})}{3(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6})}$

To find:

The quotient and write it in rectangular form using exact values

Recall the below;

$\cos\theta+i\sin\theta=e^{i\theta}$

So we can go ahead and rewrite the given expression and simplify as shown below;

$\begin{gathered} \frac{8(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2})}{3(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6})} \\ =\frac{8(e^{\frac{i\pi}{2}})}{3(e^{\frac{i\pi}{6}})} \\ =\frac{8}{3}(e^{\frac{i\pi}{2}-\frac{i\pi}{6}}) \\ =\frac{8}{3}(e^{i\pi(\frac{1}{2}-\frac{1}{6})} \\ =\frac{8}{3}e^{\frac{i\pi}{3}} \end{gathered}$

So we'll have;

$\begin{gathered} \frac{8}{3}(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}) \\ =\frac{8}{3}(\frac{1}{2}+i\frac{\sqrt{3}}{2}) \\ =\frac{8}{6}+i\frac{8\sqrt{3}}{6} \\ =\frac{4}{3}+\frac{i4\sqrt{3}}{3} \end{gathered}$