Question

How do you convert \[x=3\] to polar form?

Answer 1

Hint: In this problem, we have to convert the given rectangular form to the polar form. We know that the rectangular coordinate can be converted into its polar coordinate using the conversion formula. We can substitute the value of x in the conversion formula to get the required polar coordinate.

Complete step by step solution:
We know that the given rectangular form is,
\[x=3\]
We have to convert the given rectangular form to its polar form.
We can now use the conversion formula to convert the rectangular form to its polar form.
The conversion formula is to convert the rectangular form to its polar form.
\[\begin{align}
  & x=r\cos \theta \\
 & y=r\sin \theta \\
\end{align}\]
Here we are given only the x value, so we can use the first formula.
We can substitute the value of x in the above formula, we get
\[\Rightarrow 3=r\cos \theta \]
We can now divide by \[\cos \theta \] on both the left-hand side and the right-hand side, we get
\[\Rightarrow r=\dfrac{3}{\cos \theta }\] …… (1)
We know that \[\dfrac{1}{\cos \theta }=\sec \theta \].
We can now substitute the above value in (1), we get
\[\Rightarrow r=3\sec \theta \]
Therefore, the polar form of the rectangular coordinate \[x=3\] is \[r=3\sec \theta \].

Note: Students make mistakes while writing the conversion formula to convert from polar form to a rectangular form, we should know that the polar coordinates are given as \[\left( r,\theta \right)\] format and the rectangular coordinates are in the format \[\left( x,y \right)\]. We should remember some trigonometric identities like \[\dfrac{1}{\cos \theta }=\sec \theta \] to be used in these types of problems.