Question

What is the Cartesian form of `( -1, (4pi)/3 )`?

Answer 1

`(1/2, sqrt(3)/2)`

Explanation:

We are given the polar form, so there is a radius and an angle. We want to convert to `x,y` coordinates.

So we can use Euler's formula (or at least the idea behind it) to convert between Cartesian and polar:

`x = r costheta`
`y = r sintheta`

From that, we just plug in the numbers, remembering our unit circle:

`cos((4pi)/3) = -1/2 and sin((4pi)/3) = - sqrt(3)/2`
therefore
`(x,y) = (1/2, sqrt(3)/2)`
You could also notice that a negative radius is the same as adding or subtracting `pi` to the angle, hence
`(-1, (4pi)/3) = (1, pi/3)`
which I think is a bit easier to think about.

Answer 2

`(1/2, \sqrt3/2)`

Explanation:

The Cartesian coordinates `(x, y)` of the point `(-1, {4\pi}/3)\equiv(r, \theta)` are given as follows

`x=r\cos\theta`

`=-1\cos({4\pi}/3)`

`=-\cos(\pi+\pi/3)`

`=\cos(\pi/3)`

`=1/2`

`y=r\sin\theta`

`=-1\sin({4\pi}/3)`

`=-\sin(\pi+\pi/3)`

`=\sin(\pi/3)`

`=\sqrt3/2`

hence, the Cartesian coordinates are `(1/2, \sqrt3/2)`