Question

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

Short answer

The polar co-ordinates are$(3\sqrt{2},1.5574)and(-3\sqrt{2},-1.5574)$

Step-by-step solution

Step 1. Value of r

The co-ordinates are given as $(-3,3)$

We know,$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{3^2+{(-3)}^2} \\ r=3\sqrt{2} \end{gathered}$$

Step 2. Value of θ

The value of $\theta$can be found by,

$$\begin{gathered} \theta ={\tan }^{-1}(y/x) \\ ={\tan }^{-1}(-1/1) \\ =1.5574 \end{gathered}$$

Step 3. Value of angle with negative r

When $r=-3\sqrt{2}$, the ray will go in the opposite direction,

Therefore$\theta =-1.5574$