Question

The polar coordinates of a point are given. Find rectangular coordinates of each point.

$\left(4,\frac{3\mathrm{\pi }}{2}\right)$

Short answer

The rectangular coordinate is$\left(0,-4\right)$.

Step-by-step solution

Step 1. Given information.

The given polar coordinates of a point is$\left(4,\frac{3\mathrm{\pi }}{2}\right)$

Step 2. Finding rectangular x-coordinate.

We know that if a point $P$ on the terminal side of $\theta$and has polar coordinates $\left(r,\theta \right)$, then the corresponding rectangular coordinates are given by $x=r\cos \theta$and$y=r\sin \theta$.
On substituting we get,

$$\begin{gathered} x=r\cos \theta \\ =4\cos \frac{3\mathrm{\pi }}{2} \\ =4\left(0\right) \\ =0 \end{gathered}$$

Step 3. Finding rectangular y-coordinate.

Similarly on substituting we get,

$$\begin{gathered} y=r\sin \theta \\ =4\sin \frac{3\mathrm{\pi }}{2} \\ =4\left(-1\right) \\ =-4 \end{gathered}$$

Therefore, the rectangular coordinates are$\left(0,-4\right)$.