Question

How do you convert the rectangular equation $x=4$ into polar form?

Answer 1

We are given a rectangular equation $x=4$and we are to convert into polar coordinate $(r,\theta )$. Polar coordinates have x coordinate as $x=r\cos \theta$ and the y coordinate as $y=r\sin \theta$. We will compare the given equation $x=4$ with the polar x coordinate $x=r\cos \theta$. And we will obtain the equivalent polar coordinate in terms of $r$.

Complete step by step solution:
According to the given question, we have a rectangular equation $x=4$ which we have to write in the polar form.
Polar coordinates which is of the form $(r,\theta )$, has the x- coordinate as $x=r\cos \theta$ and the y-coordinate as $y=r\sin \theta$.
The given rectangular equation $x=4$ is a straight line parallel to the y-axis, with no y-coordinate.
So, we will now compare the given rectangular equation with the polar coordinate.
We have,
$x=4$ and $x=r\cos \theta$
On comparing the above equations, we can write the new expression as,
$4=r\cos \theta$
On rearranging the above equation, we get it as,
$\Rightarrow r=\dfrac{4}{\cos \theta }$
Therefore, the polar form of the rectangular equation $x=4$ is $r=\dfrac{4}{\cos \theta }$.

Note: In order to convert a rectangular coordinate into polar coordinate, we will be using the formula,
${{r}^{2}}={{x}^{2}}+{{y}^{2}}$
where $x=r\cos \theta$ and $y=r\sin \theta$
and the angle $\theta$ can be found by using the given values in the rectangular equation form. For finding the angle $\theta$, we will be using the formula,
$\tan \theta =\dfrac{y}{x}$
We will get the polar coordinates as $(r,\theta )$.
We can use the same steps to convert the polar coordinates back into the rectangular coordinates.
We can simply substitute all the values that we know, that is, the value of $r$ and the angle $\theta$ in the equations $x=r\cos \theta$ and $y=r\sin \theta$, and we will get the coordinates as $(x,y)$.