Question

How do you convert the rectangular equation \[x=4\] into polar form?

Answer 1

Hint: 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 \[(x,y)\].