Question

The rectangular coordinates of a point are given. Find two pairs of polar coordinates $(r,\theta )$for each point, one with $r>0$and the other with $r<0$. Express $\theta$ in radians

Short answer

The polar co-ordinates are$(2,1.571)$and$(-2,-1.571)$.

Step-by-step solution

Step 1. Value of r

The given co-ordinate is $(0,-2)$.

We know,

$r=\sqrt{x^2+y^2}$

$$\begin{gathered} r=\sqrt{0^2+(-2{)}^2} \\ =\sqrt{0+4} \\ =\sqrt{4} \\ =2 \end{gathered}$$

Step 2. Value of θ

Since the point $(0,-2)$lies on the negative Y-Axis, therefore$$\begin{gathered} \theta =-\frac{\pi }{2} \\ =1.571radians \end{gathered}$$

Step 3. Polar co-ordinate with negative value of r

When $r=-2,$

$\theta$will lie on the positive Y-Axis

Therefore, the polar co-ordinate will be$(-2,-\frac{\pi }{2})=(-2,-1.751rad)$