# What is the trigonometric form of  (2-25i) ?

Dec 17, 2016

$\sqrt{629} \left(\cos \left(1.49\right) - \sin \left(1.49\right)\right)$

#### Explanation:

The trigonometric form of a complex number $z = x + i y$

is $\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{z = r \left(\cos \theta + i \sin \theta\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

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

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\theta = {\tan}^{-} 1 \left(\frac{y}{x}\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where $- \pi < \theta \le \pi$

Here $x = 2 \text{ and } y = - 25$

$\Rightarrow r = \sqrt{{2}^{2} + {\left(- 25\right)}^{2}} = \sqrt{629}$

Now 2 - 25i is in the 4th quadrant, so we must ensure that $\theta$ is in the 4th quadrant.

$\Rightarrow \theta = {\tan}^{-} 1 \left(- \frac{25}{2}\right) = - 1.49 \leftarrow \text{ in 4th quadrant}$

$\Rightarrow 2 - 25 i = \sqrt{629} \left(\cos \left(- 1.49\right) + i \sin \left(- 1.49\right)\right)$

which can also be expressed as.

$2 - 25 i = \sqrt{629} \left(\cos \left(1.49\right) - i \sin \left(1.49\right)\right)$