Question

The polar coordinates of a point are given. Find the rectangular coordinates of this point.  What are the rectangular coordinates of this point?

Answer 1

The polar coordinates and the cartesian coordinates are related by the following relation

`\begin{cases}x=r\cos \theta \\ y=r\sin \theta\end{cases}`

We just need to substitute our values on this expression, and evaluate to find our coordinates on rectangular coordinates.

`\begin{gathered} (r,\theta)=(-5,(3\pi)/(2)) \\ \Rightarrow\begin{cases}x=-5\cos (3\pi)/(2) \\ y=-5\sin (3\pi)/(2)\end{cases} \end{gathered}`

Those trigonometric functions have known values

`\begin{gathered} \cos (3\pi)/(2)=0 \\ \sin (3\pi)/(2)=-1 \end{gathered}`

Plugging those values in our expression, we have

`\begin{cases}x=-5\cdot(0) \\ y=-5\cdot(-1)\end{cases}\Rightarrow\begin{cases}x=0 \\ y=5\end{cases}`

Our point written in rectangular coordinates is

`(0,5)`