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 $-\frac{27}{2}-\frac{27\sqrt{3}}{2}i$.

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 $z=\left(a+bi\right)$,

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

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

Using the equation (i),

$$\begin{gathered} z^3=3^3\left[\cos \left(3·80°\right)+i\sin \left(3·80°\right)\right] \\ {z}^3=27\left(\cos 240°+i\sin 240°\right) \\ {z}^3=27\left[-\frac{1}{2}+i\left(\frac{-\sqrt{3}}{2}\right)\right] \\ {z}^3=-\frac{27}{2}-\frac{27\sqrt{3}}{2}i \end{gathered}$$