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.