Question

What is the Cartesian form of `(36,pi)`?

Answer 1

`(-36,0)`

Explanation:

The formulas used to convert between polar and Cartesian co-ordinates are:
`x=rcostheta`
`y=rsintheta`

We are given the polar co-ordinate point `(r,theta)->(36,pi)`, so `r=36` and `theta=pi`. Making substitutions in the above formulas:
`x=36cos(pi)=36*(-1)=-36`
`y=36sin(pi)=36(0)=0`

Therefore, the point in the Cartesian plane is `(-36,0)`.

Intuitively, this result makes sense because `pi` is a half-circle rotation, so in the Cartesian plane an angle of `pi` corresponds to a point on the `x`-axis (and therefore `y=0`). A radius of `36` means the point is `36` units to the left or right of the origin, and since an angle of `+pi` is a counterclockwise rotation, it would mean the point is `36` units to the left (and therefore `-36`).