Question

What is the Cartesian form of `(12,(5pi )/3)`?

Answer 1

`(6,-6sqrt(3))`

Explanation:

Remember that polar coordinates are of the form `(r, theta)`.

Also, remember that our `x`-values correspond with cosine and `y`-values with sine.

Then, remember that our sine and cosine values come from the unit circle, where `r=1`. So, when changing our coordinates from polar to cartesian coordinates we are taking `(r, theta) -> (rcos(theta), rsin(theta))`.

Notice that `(5pi)/3=2pi-pi/3`. So, we can say that `theta=-pi/3`, which is in the fourth quadrant. This means that cosine is positive, and sine is negative.

Then we can essentially say that `(12, (5pi)/3) -> (12cos(pi/3),-12sin(pi/3))`
`=(12(1/2),-12((sqrt(3))/2))=(6,-6sqrt(3))`