# How do you divide ( 2i+5) / ( -7 i + 7 ) in trigonometric form?

Feb 11, 2018

$0.54 \left(\cos \left(1.17\right) + i \sin \left(1.17\right)\right)$

#### Explanation:

Let's split them up into two separate complex numbers to start with, one being the numerator, $2 i + 5$, and one the denominator, $- 7 i + 7$.

We want to get them from linear ($x + i y$) form to trigonometric ($r \left(\cos \theta + i \sin \theta\right)$ where $\theta$ is the argument and $r$ is the modulus.

For $2 i + 5$ we get

$r = \sqrt{{2}^{2} + {5}^{2}} = \sqrt{29}$

$\tan \theta = \frac{2}{5} \to \theta = \arctan \left(\frac{2}{5}\right) = 0.38 \text{ rad}$

and for $- 7 i + 7$ we get

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

Working out the argument for the second one is more difficult, because it has to be between $- \pi$ and $\pi$. We know that $- 7 i + 7$ must be in the fourth quadrant, so it will have a negative value from $- \frac{\pi}{2} < \theta < 0$.

That means we can figure it out simply by

$- \tan \left(\theta\right) = \frac{7}{7} = 1 \to \theta = \arctan \left(- 1\right) = - 0.79 \text{ rad}$

So now we've got the complex number overall of

$\frac{2 i + 5}{- 7 i + 7} = \frac{\sqrt{29} \left(\cos \left(0.38\right) + i \sin \left(0.38\right)\right)}{7 \sqrt{2} \left(\cos \left(- 0.79\right) + i \sin \left(- 0.79\right)\right)}$

We know that when we have trigonometric forms, we divide the moduli and subtract the arguments, so we end up with

$z = \left(\frac{\sqrt{29}}{7 \sqrt{2}}\right) \left(\cos \left(0.38 + 0.79\right) + i \sin \left(0.38 + 0.79\right)\right)$

$= 0.54 \left(\cos \left(1.17\right) + i \sin \left(1.17\right)\right)$