Q 54. - Calculus

Question

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

$y=x+1$

Step-by-Step Solution

Answer

The required equation is $r=\frac{1}{\left(\sin \theta -\cos \theta \right)}$.

Step 1. Given information.

The given equation in rectangular coordinates is:

$y=x+1$

Step 2. Find the equation in rectangular coordinates.

$y=x+1$

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,

$$\begin{gathered} r \sin \theta =r \cos \theta +1 \\ r \sin \theta -r \cos \theta =1 \\ r\left(\sin \theta -\cos \theta \right)=1 \end{gathered}$$

Divide both sides by $\sin \theta -\cos \theta$,

$$\begin{gathered} \frac{r\left(\sin \theta -\cos \theta \right)}{\sin \theta -\cos \theta }=\frac{1}{\sin \theta -\cos \theta } \\ r=\frac{1}{\sin \theta -\cos \theta } \end{gathered}$$

Therefore, the equation in polar coordinates is $r=\frac{1}{\left(\sin \theta -\cos \theta \right)}$.