Question

How do you find the rectangular equation for $\theta = \dfrac{{5\pi }}{6}$ ?

Answer 1

Hint : In this question, we are given the value of $\theta$ , so we are given the polar form of the equation and we have to convert it into a rectangular equation, it means that we have to express y in terms of x. We know that $x = r\cos \theta$ and $y = r\sin \theta$ , using the value of $\theta$ we will find the value of both x and y in terms of r and thus get the value of y in terms of x.

Complete step by step solution:
We have to find the rectangular form of $\theta = \dfrac{{5\pi }}{6}$
Let a right-angled triangle be 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 trigonometry we have –
$\cos \theta = \dfrac{{base}}{{hypotenuse}} = \dfrac{x}{{\sqrt {{x^2} + {y^2}} }} = \dfrac{x}{r} \\\ \Rightarrow x = r\cos \theta \;$
And similarly $y = r\sin \theta$
As we know the value of $\theta$ , we can find the value of x and y.
$x = r\cos \theta \\\ \Rightarrow x = r\cos \dfrac{{5\pi }}{6} \\\ \Rightarrow x = r\cos (\pi - \dfrac{\pi }{6}) = r( - \cos \dfrac{\pi }{6}) = - r\cos \dfrac{\pi }{6} \\\ y = r\sin \dfrac{{5\pi }}{6} = r\sin (\pi - \dfrac{\pi }{6}) = r\sin \dfrac{\pi }{6} \\\$
Now,
$\dfrac{y}{x} = \dfrac{{r\sin \dfrac{\pi }{6}}}{{ - r\cos \dfrac{\pi }{6}}} = - \tan \dfrac{\pi }{6} \\\ \Rightarrow \dfrac{y}{x} = \dfrac{{ - 1}}{{\sqrt 3 }} \\\ \Rightarrow y = \dfrac{{ - x}}{{\sqrt 3 }} \;$
Hence the rectangular form of $\theta = \dfrac{{5\pi }}{6}$ is $y = \dfrac{{ - x}}{{\sqrt 3 }}$ .
So, the correct answer is “ $\theta = \dfrac{{5\pi }}{6}$ is $y = \dfrac{{ - x}}{{\sqrt 3 }}$ ”.

Note : There are two types of coordinates for plotting a point on the graph paper namely rectangular coordinate system and polar coordinate system. The rectangular coordinate system is the most commonly used coordinate system and 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 polar coordinate system is of the form $(r,\theta )$ 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.