Question

In Exercises 48–55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$y=mx$

Short answer

The required equation is$\tan \theta =m$.

Step-by-step solution

Step 1. Given information.

The given equation in rectangular coordinates is:

$y=mx$

Step 2. Find the equation in rectangular coordinates.

$y=mx$

As we know that in polar coordinates $x=r\cos \theta \mathrm{and}y=r\sin \theta$, then substitute the value of yand x,

$r\sin \theta =m·r\cos \theta$

Divide both sides by r,

$$\begin{gathered} \frac{r\sin \theta }{r}=\frac{m·r\cos \theta }{r} \\ \sin \theta =m\cos \theta \end{gathered}$$

Divide both sides by role="math" localid="1649352786439" $\cos \theta$,

$$\begin{gathered} \frac{\sin \theta }{\cos \theta }=\frac{m\cos \theta }{\cos \theta } \\ \tan \theta =m \end{gathered}$$

Therefore, the equation in polar coordinates is$\tan \theta =m$.