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$(5,0.93)$and$(-5,4.07)$

Step-by-step solution

Step 1. Value of r

The given rectangular co-ordinate is $(3,4)$

We know,

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{3^2+4^2} \\ =\sqrt{9+16} \\ =\sqrt{25} \\ =5 \end{gathered}$$

Step 2. Value of θ

Now, the angle $\theta$can be found by

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ \theta ={\tan }^{-1}\left(\frac{4}{3}\right) \\ ={\tan }^{-1}(1.33) \\ =0.93 \end{gathered}$$

So the point becomes,$(5,0.93)$

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

Now,

when $r=-5$$r=-5$

The ray will go in the opposite direction,

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{y}{x}\right),-\frac{\pi }{2}<\theta <\frac{\pi }{2} \\ \theta ={\tan }^{-1}\left(\frac{4}{3}\right) \\ ={\tan }^{-1}(1.33) \\ =-4.07 \end{gathered}$$