Q 55. - Calculus

Question

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

$y=mx$

Step-by-Step Solution

Answer

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

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 y and 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 $\cos \theta$,

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

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