Question

Write the expression in the standard form $a+ib$.

${\left[3\left(\cos 80°+i\sin 80°\right)\right]}^3$

Short answer

The expression in the standard form is $-2\sqrt{2}-2i\sqrt{2}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

${\left[3\left(\cos 80°+i\sin 80°\right)\right]}^3$

We know that for a complex number raise to n when it is in the polar form $z^n=r^n\left(\cos \theta +i\sin \left(n\theta \right)\right)......\left(i\right)$

Write the expression in the standard form $a+ib$,

$z^4={\left[\sqrt{2}\left(\cos \frac{5\mathrm{\pi }}{16}+i\sin \frac{5\mathrm{\pi }}{16}\right)\right]}^4$

Step 2. Use equation (i) and solve the equation.

Using the equation (i),

$$\begin{gathered} z^4=\cos \left(4·\frac{5\mathrm{\pi }}{16}\right)+i\sin \left(4·\frac{5\mathrm{\pi }}{16}\right) \\ {z}^4=4\left(\cos \frac{5\mathrm{\pi }}{4}+i\sin \frac{5\mathrm{\pi }}{4}\right) \\ {z}^4=4\left[-\frac{\sqrt{2}}{2}+i\left(-\frac{\sqrt{2}}{2}\right)\right] \\ {z}^4=-2\sqrt{2}-2i\sqrt{2} \end{gathered}$$