# How do you write the complex number in trigonometric form 1+3i?

Sep 18, 2016

$\sqrt{10} \left(\cos \left(1.249\right) + i \sin \left(1.249\right)\right)$

#### Explanation:

To convert from $\textcolor{b l u e}{\text{complex to trigonometric form}}$

That is $x + y i \to r \left(\cos \theta + i \sin \theta\right)$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{r = \sqrt{{x}^{2} + {y}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

and $\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\theta = {\tan}^{-} 1 \left(\frac{y}{x}\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}} \left(- \pi < \theta \le \pi\right)$

here x = 1 and y = 3

$\Rightarrow r = \sqrt{{1}^{2} + {3}^{2}} = \sqrt{10}$

Now 1 + 3i is in the 1st quadrant so $\theta$ must be in the 1st quadrant.

$\theta = {\tan}^{-} 1 \left(3\right) = 1.249 \text{ rad" larr" in 1st quad.}$

$\Rightarrow 1 + 3 i \to \sqrt{10} \left(\cos \left(1.249\right) + i \sin \left(1.249\right)\right)$