# What is the trigonometric form of  (-7+2i) ?

Mar 4, 2017

$\sqrt{53} \left(\cos \left(2.86\right) + i \sin \left(2.86\right)\right)$

#### Explanation:

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

$\text{that is } \left(x , y\right) \to r \left(\cos \theta + i \sin \theta\right)$

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

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

$\text{here " x=-7" and } y = 2$

$\Rightarrow r = \sqrt{{\left(- 7\right)}^{2} + {2}^{2}} = \sqrt{53}$

$\text{Since } - 7 + 2 i$ is in the second quadrant, we must ensure that $\theta$ is in the second quadrant.

$\theta = \pi - {\tan}^{-} 1 \left(\frac{2}{7}\right)$

$\Rightarrow \theta = \left(\pi - 0.278\right) = 2.86 \leftarrow \text{ in second quadrant}$

$\Rightarrow - 7 + 2 i \to \sqrt{53} \left(\cos \left(2.86\right) + i \sin \left(2.86\right)\right)$