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

Apr 10, 2018

color(blue)(sqrt(37)[cos(2.976)+isin(2.976)]

#### Explanation:

The trigonometric for of a complex number $\left(a + b i\right)$ is given by:

$z = r \left[\cos \left(\theta\right) + i \sin \left(\theta\right)\right]$

Where:

$r = \sqrt{{a}^{2} + {b}^{2}}$

$\theta = \arctan \left(\frac{b}{a}\right)$

$r = \sqrt{{\left(- 6\right)}^{2} + {\left(1\right)}^{2}} = \sqrt{37}$

$\theta = \arctan \left(\frac{1}{-} 6\right) \approx - 0.1651486774$

This is in the IV quadrant.

Remember that $\tan \left(\theta\right)$ only has an inverse for the domain:

$- \frac{\pi}{2} < \theta < \frac{\pi}{2}$

So $\arctan \left(y\right)$ will return angles in this range.

So we need to add $\pi$ to this result, since $\left(- 6 + i\right)$ is in the II quadrant:

$- 0.1651486774 + \pi = 2.976$ 3 d.p.

So:

$\theta = 2.976$

Trigonometric form is therefore:

color(blue)(sqrt(37)[cos(2.976)+isin(2.976)]