Chapter I, Problem 46E

To determine

To write: The number $e^{\pi +i}$ in the form $a+bi$.

Answer to Problem 46E

The number $e^{\pi +i}$ expressed in the form of complex number as $e^{\pi }\cos 1+\left(e^{\pi }\sin 1\right)i$.

Explanation of Solution

Formula used:

Euler’s formula:

$e^{iy}=\cos y+i\sin y$

Calculation:

Simplify the number $e^{\pi +i}$ by using Euler’s formula as follows.

$\begin{array}{c}{e}^{\pi +i}=e^{\pi }{e}^i\\ =e^{\pi }\left(\cos 1+i\sin 1\right)\\ =e^{\pi }\cos 1+\left(e^{\pi }\sin 1\right)i\end{array}$

Thus, the number $e^{\pi +i}$ expressed in the form of complex number as $e^{\pi }\cos 1+\left(e^{\pi }\sin 1\right)i$.