Q 52. - Calculus

Question

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

$y=\sqrt{3}x$

Step-by-Step Solution

Answer

The required equation is $\theta =\frac{\mathrm{\pi }}{3}$.

Step 1. Given information.

The given equation in rectangular coordinates is:

$y=\sqrt{3}x$

Step 2. Find the equation in rectangular coordinates.

$y=\sqrt{3}x$

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

$r \sin \theta =\sqrt{3}·r \cos \theta$

Divide both sides by r,

$$\begin{gathered} \frac{r \sin \theta }{r}=\frac{\sqrt{3}·r \cos \theta }{r} \\ \sin \theta =\sqrt{3}\cos \theta \end{gathered}$$

Divide both sides by $\cos \theta$,

$$\begin{gathered} \frac{\sin \theta }{\cos \theta }=\frac{\sqrt{3}\cos \theta }{\cos \theta } \\ \tan \theta =\sqrt{3} \\ \theta =\frac{\mathrm{\pi }}{3} \end{gathered}$$

Therefore, the equation in polar coordinates is $\theta =\frac{\mathrm{\pi }}{3}$.