# How do you divide ( -i-5) / (i +4 ) in trigonometric form?

Jul 13, 2017

$1.24$ $c i s$ $3.09$

#### Explanation:

First, divide the expression so that you can get a complex number in $a + b i$ form. To do this, you must multiply the numerator and denominator by the conjugate, $i - 4$.

$\frac{- i - 5}{i + 4}$

$= \frac{- i - 5}{i + 4} \cdot \frac{i - 4}{i - 4}$ $\to$multiply by the conjugate

$= \frac{- {i}^{2} + 4 i - 5 i + 20}{{i}^{2} - {4}^{2}}$ $\to$expand

$= \frac{1 + 4 i - 5 i + 20}{- 1 - 16}$ $\to$simplify

$= \frac{21 - i}{- 17}$ $\to$combine like terms

$= \textcolor{b l u e}{- \frac{21}{17} + \frac{1}{17} i}$ $\to$rewrite in $a + b i$ form

To convert this to trigonometric form, you must find out $r$, the distance from the origin to the point, and $\theta$, the angle. Use the following formulas:

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

$\tan \theta = \frac{b}{a}$

In this case, $a = - \frac{21}{17}$ and $b = \frac{1}{17}$.

$r = \sqrt{{a}^{2} + {b}^{2}} = \sqrt{{\left(- \frac{21}{17}\right)}^{2} + {\left(\frac{1}{17}\right)}^{2}} \approx \textcolor{b l u e}{1.24}$
$\tan \theta = \frac{\frac{1}{17}}{- \frac{21}{17}} = \frac{1}{17} \cdot - \frac{17}{21} = - \frac{1}{21}$

$\theta = {\tan}^{-} 1 \left(- \frac{1}{21}\right) \approx - 0.05$

However, since the coordinate $\left(- \frac{21}{17} + \frac{1}{17} i\right)$ is in Quadrant $I I$, this angle is wrong. This is because we used the $\arctan$ function, which doesn't account for angles outside of the range $\left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$. To fix this, add $\pi$ to $\theta$.

$- 0.05 + \pi = \textcolor{b l u e}{3.09}$

So, the trigonometric form is $1.24$ $c i s$ $3.09$, or $1.24 \left(\cos 3.09 + i \sin 3.09\right)$.