# What is the trigonometric form of  (1-3i) ?

May 2, 2016

$\sqrt{10} \left(\cos \left(- 1.25\right) + i \sin \left(- 1.25\right)\right)$

#### Explanation:

Given a complex number z = x + iy , then in trig.form it is written

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

where $| z | = | x + i y | = r = \sqrt{{x}^{2} + {y}^{2}}$

and $\theta = {\tan}^{-} 1 \left(\frac{y}{x}\right)$

here x = 1 and y = - 3

$\Rightarrow r = \sqrt{{1}^{2} + {\left(- 3\right)}^{2}} = \sqrt{10}$

and $\theta = {\tan}^{-} 1 \left(- 3\right) = - 1.25 \text{ radians }$

$\Rightarrow \left(1 - 3 i\right) = \sqrt{10} \left(\cos \left(- 1.25\right) + i \sin \left(- 1.25\right)\right)$