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# What is the trigonometric form of  (-6+8i) ?

Jun 15, 2018

$10 \left(\cos {126.87}^{\circ} + i \sin {126.87}^{\circ}\right)$

#### Explanation:

$\text{the trigonometric form of a complex number is}$

•color(white)(x)r(costheta+isintheta)" where"

•color(white)(x)r=sqrt(x^2+y^2)

•color(white)(x)theta=tan^-1(y/x)

$- 6 + 8 i \to x = - 6 \text{ and } y = 8$

$r = \sqrt{{\left(- 6\right)}^{2} + {8}^{2}} = \sqrt{100} = 10$

$- 6 + 8 i \text{ is in the second quadrant so "theta" must be}$
$\text{an angle in the second quadrant}$

$\theta = {\tan}^{-} 1 \left(\frac{4}{3}\right) = {53.13}^{\circ} \leftarrow \textcolor{red}{\text{related acute angle}}$

$\theta = {\left(180 - 53.13\right)}^{\circ} = {126.87}^{\circ} \leftarrow \textcolor{red}{\text{in second quadrant}}$

$- 6 + 8 i = 10 \left(\cos {126.87}^{\circ} + i \sin {126.87}^{\circ}\right)$

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