# What is the trigonometric form of  (-5+12i) ?

Jan 4, 2016

13 ( cos1.96 +isin1.96)

#### Explanation:

the complex number z = x + iy can be written in trigonemetric form as $z = r \left(\cos \theta + i \sin \theta\right)$

the modulus r =$\sqrt{{x}^{2} + {y}^{2}}$

and the argument $\theta$ is found by using

$\tan \theta = \frac{y}{x}$

in this case $r = \sqrt{{\left(- 5\right)}^{2} + {\left(12\right)}^{2}}$

$= \sqrt{25 + 144} = \sqrt{169} = 13$

$\tan \alpha = \frac{y}{x} = \frac{12}{5} = 2.4$ and so

$\alpha = 1. 176$

arg z = $\left(\pi - 1.176\right)$ = 1.96

In this case z is in the 2nd quadrant and so the required argument is $\left(\pi - \alpha\right)$