# How do you write the complex number in trigonometric form 6-7i?

Sep 3, 2017

$\sqrt{85} \left(\cos \left(0.862\right) - i \sin \left(0.862\right)\right)$

#### Explanation:

$\text{to convert from"color(blue)" complex to trig. form}$

$\text{that is "x+yitor(costheta+isintheta)" using}$

•color(white)(x)r=sqrt(x^2+y^2)

•color(white)(x)theta=tan^-1(y/x)color(white)(x);-pi< theta <=pi

$\text{here "x=6" and } y = - 7$

$\Rightarrow r = \sqrt{{6}^{2} + {\left(- 7\right)}^{2}} = \sqrt{85}$

$6 - 7 i \text{ is in the fourth quadrant so we must ensure that } \theta$
$\text{is in the fourth quadrant}$

$\Rightarrow \theta = {\tan}^{-} 1 \left(\frac{7}{6}\right) = 0.862 \leftarrow \textcolor{red}{\text{ related acute angle}}$

$\Rightarrow \theta = - 0.862 \leftarrow \textcolor{red}{\text{ in fourth quadrant}}$

$\Rightarrow 6 - 7 i = \sqrt{85} \left(\cos \left(- 0.862\right) + i \sin \left(- 0.862\right)\right)$

[cos(-0.862=cos(0.862);sin(-0.862)=-sin(0.862)]

$\Rightarrow 6 - 7 i = \sqrt{85} \left(\cos \left(0.862\right) - i \sin \left(0.862\right)\right)$