# What is the trigonometric form of  (-2+9i) ?

Jan 1, 2016

$\sqrt{85} \left(\cos \left({\tan}^{-} 1 \left(- \frac{9}{2}\right)\right) + i \sin \left({\tan}^{-} 1 \left(- \frac{9}{2}\right)\right)\right)$

#### Explanation:

$\left(- 2 + 9 i\right)$

$r \cos \left(\theta\right) = - 2$
$r \sin \left(t h e a\right) = 9$

${r}^{2} {\cos}^{2} \left(\theta\right) = 4$
${r}^{2} {\sin}^{2} \left(\theta\right) = 81$

${r}^{2} {\cos}^{2} \left(\theta\right) + {r}^{2} {\sin}^{2} \left(\theta\right) = 4 + 81$
${r}^{2} \left({\cos}^{2} \left(\theta\right) + {\sin}^{2} \left(\theta\right)\right) = 85$
${r}^{2} = 85$
$r = \sqrt{85}$

$r \sin \frac{\theta}{r} \cos \left(\theta\right) = \frac{9}{-} 2$
$\tan \left(\theta\right) = - \frac{9}{2}$

$\theta = {\tan}^{-} 1 \left(- \frac{9}{2}\right)$

The complex number in trigonometric form is

$\sqrt{85} \left(\cos \left({\tan}^{-} 1 \left(- \frac{9}{2}\right)\right) + i \sin \left({\tan}^{-} 1 \left(- \frac{9}{2}\right)\right)\right)$