Question

In Problems 1–3, plot each point given in polar coordinates, and find its rectangular coordinates:

$\left(3,\frac{\pi }{6}\right)$

Short answer

The rectangular co-ordinates of the polar co-ordinate $\left(3,\frac{\pi }{6}\right)$are$\left(\frac{3\sqrt{3}}{2},\frac{3}{2}\right)$and the graph is

Step-by-step solution

Step 1. Given Information

We are given the polar co-ordinates as$\left(3,\frac{\pi }{6}\right)$

Comparing it with standard polar co-ordinates$(r,\theta )$, we get,

$$\begin{gathered} r=3 \\ \theta =\frac{\pi }{6} \end{gathered}$$

Step 2. Plotting the polar co-ordinate

Since the angle is $\frac{\pi }{6}$, we will make the angle between the axis and the line as $\frac{\pi }{6}$. Also the length is given as $r=3$

Therefore, the graph will look like,

Step 3. Rectangular co-ordinates.

We know, the relation between polar and rectangular co-ordinates is,

$x=r\cos \theta \text{and}y=r\sin \theta$

Thus, the value of

$$\begin{gathered} x=3·\cos \frac{\pi }{6} \\ =\frac{3\sqrt{3}}{2} \end{gathered}$$

The value of $$\begin{gathered} y=r\sin \theta \\ =3·\sin \frac{\pi }{6} \\ =\frac{3}{2} \end{gathered}$$

The co-ordinate will be$\left(\frac{3\sqrt{3}}{2},\frac{3}{2}\right)$