Chapter 14.4, Problem 3OE

To determine

To find: The given complex number in $x+yi$ form.

Answer to Problem 3OE

  $1+i$ .

Explanation of Solution

Given information:

A complex number in polar form is given as $\sqrt{2}\left(\cos {45}^0+i\sin {45}^0\right)$ .

Concepts used: The polar form of a complex number $z=x+yi$ is given by −

  $z=r\left(\cos \theta +i\sin \theta \right)$

where $r=\sqrt{x^2+y^2},\cos \theta =\frac{x}{r},\sin \theta =\frac{y}{r}$ .

Calculation:

Given complex number in polar form is $\sqrt{2}\left(\cos {45}^0+i\sin {45}^0\right)$ .

Converting into $x+yi$ form -

  $\begin{array}{l}\sqrt{2}\left(\cos {45}^0+i\sin {45}^0\right)\\ =\sqrt{2}\left(\frac{\sqrt{2}}{2}+i\times \frac{\sqrt{2}}{2}\right)\left\{{\text{Since, cos45}}^0=\frac{\sqrt{2}}{2},\sin {45}^0=\frac{\sqrt{2}}{2}\right\}\\ =1+i\end{array}$

Hence, given complex number in desired form is $1+i$ .