Question

Write each expression in the standard form for a complex number, a + bi.
a. [3(cos(27°)) + isin(27°)]^5
b. [2(cos(40°)) + isin(40°)]^6

Answer 1

\[\large\begin{align*}\bigg[3(\cos27^\circ+i\sin27^\circ)\bigg]^5&=\left(3e^{27^\circ i}\right)^5\\
&=3^5e^{27\cdot5i}\\
&=3^5e^{135^\circ i}\\
&=3^5\bigg[\cos135^\circ +i\sin135^\circ\bigg]\\
&=3^5\left[-\frac{1}{\sqrt2} +\frac{1}{\sqrt2}i\right]\\
&=-\frac{243}{\sqrt2} +\frac{243}{\sqrt2}i
\end{align*}\]
(b) can be done similarly.