Question

How do you write the polar equation $\theta = \dfrac{\pi }{3}$ in rectangular form?

Answer 1

Hint: We are given a polar equation that we have to convert into a rectangular form. We will express both the left-hand side and the right-hand side of the given equation in terms of x and y so that we get all the quantities in terms of x and y and we get the rectangular form of the given polar equation by further solving the equation.

Complete step-by-step answer:
A right-angled triangle is formed by x, y and r, where r is the hypotenuse, x is the base and y is the height of the triangle, so by Pythagoras theorem, we have - ${x^2} + {y^2} = {r^2}$ and by looking at the components of r - $x = r\cos \theta \,and\,y = r\sin \theta $ and by trigonometry or by dividing the components we have - $\tan \theta = \dfrac{y}{x}$ .
We are given $\theta = \dfrac{\pi }{3}$
We know –
$
  \tan \theta = \dfrac{y}{x} \\
   \Rightarrow \theta = {\tan ^{ - 1}}\dfrac{y}{x} \;
 $
Using this value in the given equation –
$
  {\tan ^{ - 1}}\dfrac{y}{x} = \dfrac{\pi }{3} \\
   \Rightarrow \dfrac{y}{x} = \tan \dfrac{\pi }{3} \\
   \Rightarrow \dfrac{y}{x} = \sqrt 3 \\
   \Rightarrow y = \sqrt 3 x \;
 $
Hence, the polar equation $\theta = \dfrac{\pi }{3}$ in rectangular form is $y = \sqrt 3 x$ .
So, the correct answer is “$y = \sqrt 3 x$”.

Note: The most commonly used coordinate system is the rectangular coordinate system that is also called the cartesian system and it is of the form $(x,y)$ where x is the distance of this point from the y-axis and y is the distance of the point from the x-axis. The points of the form $(r,\theta )$ are polar coordinate systems, where r is the distance of the point from the origin and $\theta $ is the counter-clockwise angle between the line joining the point and the origin, and the x-axis.