In order to convert between cartesian and polar coordinates, we have to use Pythagoras' theorem.
Consider the point (5,−2), 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 5-2i. Let's use this graph to explain how to do that: ![enter image source here]()
In order to convert 5−2i to polar form, you'll have to work out the value of r, the distance, and the size of the angle θ, the direction. Using Pythagoras' theorem, we can work that r2=52+22=29, so r=√29=5.4
Now, to work out θ, 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:
![enter image source here]()
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 5−2i.
To obtain this, though, is very simple. If where r touches the unit circle has x-coordinate cos(θ) and y coordinate sin(θ), then 5−2i has x coordinate rcos(θ) and y coordinate rsin(θ). (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 tanθ = sinθcosθ. Using our r values, we get the equation: tanθ = rsinθcosθ = −25. Therefore, θ =tan−1(−25) =−21.8
