r=5.4,θ= −21.8
Explanation:
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 5-2 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 θ , the direction. Using Pythagoras' theorem, we can work that
r 2=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 tor . (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 i.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 5−2 i has x coordinate r cos(θ) and y coordinate r sin(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: tan θ=r sinθcosθ = −25. Therefore, θ =tan−1(−25) =−21.8
