# How do you convert 5-2i to polar form?

Oct 2, 2016

$r = 5.4 , \theta = - 21.8$

#### Explanation:

In order to convert between cartesian and polar coordinates, we have to use Pythagoras' theorem.

Consider the point $\left(5 , - 2\right)$, which is the point you're trying to convert from cartesian to polar form. The Polar form gives direction and distance to any point on a graph, so that's what we'll do for 5-2$i$. Let's use this graph to explain how to do that:

In order to convert $5 - 2 i$ to polar form, you'll have to work out the value of $r$, the distance, and the size of the angle $\theta$, the direction. Using Pythagoras' theorem, we can work that ${r}^{2} = {5}^{2} + {2}^{2} = 29$, so $r = \sqrt{29} = 5.4$

Now, to work out $\theta$, we have to work out the radial angle from the positive $x$-axis to $r$. (There's a mistake in the picture; apologies about that.) Let's look at this picture now:

Using a unit circle, we can easily figure out the radial coordinates of where $r$ touches the unit circle. But this isn't what we're interested in; we want the coordinates of $5 - 2 \text{i}$.

To obtain this, though, is very simple. If where $r$ touches the unit circle has $x$-coordinate $\cos \left(\theta\right)$ and $y$ coordinate sin(theta), then $5 - 2 i$ has $x$ coordinate $r \cos \left(\theta\right)$ and $y$ coordinate $r \sin \left(\theta\right)$. (We can equate these to their cartesian coordinates, too.)

Now, to evaluate these two, we need to put them into a trigonometric function. In this case, we will use $\tan \theta$ $=$ $\sin \frac{\theta}{\cos} \theta$. Using our $r$ values, we get the equation: tan$\theta$ $=$ $r$$\sin \frac{\theta}{\cos} \theta$ = $- \frac{2}{5}$. Therefore, $\theta$ $= {\tan}^{-} 1$$\left(- \frac{2}{5}\right)$ $= - 21.8$