Question

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

$y=x$

Short answer

The required equation is$\theta =\frac{\mathrm{\pi }}{4}$.

Step-by-step solution

Step 1. Given information.

The given equation in rectangular coordinates is:

$y=x$

Step 2. Find the equation in rectangular coordinates. 

$y=x$

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

$r\cos \theta =r\sin \theta$

Divide both sides by r,

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

Divide both sides by localid="1649348673648" $\sin \theta$,

$$\begin{gathered} \frac{\cos \theta }{\sin \theta }=\frac{\sin \theta }{\sin \theta } \\ \cot \theta =1 \\ \theta =\frac{\mathrm{\pi }}{4}\left[\cot \frac{\mathrm{\pi }}{4}=1\right] \end{gathered}$$

Therefore, the equation in polar coordinates islocalid="1649348683160" $\theta =\frac{\mathrm{\pi }}{4}$.