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

`(4)/(3)+(4√(3))/(3)i`

Step-by-step explanation:

Given:

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

So we'll have;

`\begin{gathered} (8)/(3)(\cos(\pi)/(3)+i\sin(\pi)/(3)) \\ =(8)/(3)((1)/(2)+i(√(3))/(2)) \\ =(8)/(6)+i(8√(3))/(6) \\ =(4)/(3)+(i4√(3))/(3) \end{gathered}`