Question

In Problems $23-32$, write each complex number in rectangular form.

$4\left(\cos \frac{\mathrm{\pi }}{2}+i\sin \frac{\mathrm{\pi }}{2}\right)$.

Short answer

The rectangular form of the complex number$4\left(\cos \frac{\mathrm{\pi }}{2}+i\sin \frac{\mathrm{\pi }}{2}\right)$is,$4i$.

Step-by-step solution

Step 1 Given complex number is,

$4\left(\cos \frac{\mathrm{\pi }}{2}+i\sin \frac{\mathrm{\pi }}{2}\right)$.

we know that,

$$\begin{gathered} \cos \frac{\mathrm{\pi }}{2}=0 \\ \sin \frac{\mathrm{\pi }}{2}=1 \end{gathered}$$

Step 2 Substitute the known values in the given expression.

$$\begin{gathered} z=4\left(\cos \frac{\mathrm{\pi }}{2}+i\sin \frac{\mathrm{\pi }}{2}\right) \\ z=4\left(0+i\right) \\ z=4i \end{gathered}$$

Thus the required rectangular form is,$z=4i$.