Chapter I, Problem 27E

To determine

To write: The number $3+4i$ in polar form with argument between 0 and $2\pi$.

Answer to Problem 27E

The number $3+4i$ written in polar form as $3+4i=5\left[\cos \left(ta{n}^{-1}\left(\frac{4}{3}\right)\right)+i\sin \left({\tan }^{-1}\left(\frac{4}{3}\right)\right)\right]$

Explanation of Solution

The polar form of the complex number $z=a+bi$ is $z=r\left(\cos \theta +i\sin \theta \right)$ where $r=\left|z\right|=\sqrt{a^2+b^2}$  and $\tan \theta =\frac{b}{a}$.

Consider the complex number $3+4i$.

Obtain the argument of the complex number $3+4i$.

$\begin{array}{c}\tan \theta =\frac{b}{a}\\ =\frac{4}{3}\end{array}$

Thus, the argument of argument of the complex number $3+4i$ is $\theta ={\tan }^{-1}\left(\frac{4}{3}\right)$

Obtain the modulus of the complex number $3+4i$.

$\begin{array}{c}r=\left|3+4i\right|=\\ =\sqrt{3^2+4^2}\\ =\sqrt{25}\\ =5\end{array}$

Thus, the value of $r=5$.

Therefore, the polar form of the complex number $3+4i$ is $3+4i=5\left[\cos \left(ta{n}^{-1}\left(\frac{4}{3}\right)\right)+i\sin \left({\tan }^{-1}\left(\frac{4}{3}\right)\right)\right]$.