Question

How do you find rectangular coordinates for the point with polar coordinates $ \left( {4,\dfrac{{4\pi }}{3}} \right) $ ?

Answer 1

Hint: In order to find the rectangular coordinates $ \left( {x,y} \right) $ ,use the transformation
$
  x = r\cos \theta \\
  y = r\sin \theta \;
$
where r is equal to 4 and $ \theta $ is equal to $ \dfrac{{4\pi }}{3} $ to get the rectangular coordinates $ \left( {x,y} \right) $

Complete step-by-step answer:
There are two ways to determine a point on a plane, one is by the rectangular coordinates and another is by the Polar Coordinates.
Polar Coordinates $ (p,\theta ) $ is actually a 2D coordinate system in which every point on the plane is found by a distance $ p $ from a reference point and an angle i.e. $ \theta $ from a reference direction.
where $ p $ is the radial coordinate and $ \theta $ is known as the angular coordinate.
We are given a polar coordinate $ \left( {4,\dfrac{{4\pi }}{3}} \right) $
Radial coordinate = $ p\,/\,r = 4 $
Angular coordinate $ = \theta = \dfrac{{4\pi }}{3} $
Now to transformation by which we can find our rectangular coordinates $ \left( {x,y} \right) $ is
 $
  x = r\cos \theta \\
  y = r\sin \theta \;
  $
In our case $ r = 4\,and\,\theta = \dfrac{{4\pi }}{3} $
 $
  x = 4\cos \left( {\dfrac{{4\pi }}{3}} \right) \\
   = 4\cos \left( {\pi + \dfrac{\pi }{3}} \right) \;
  $
Using Allied angle in trigonometry $ \cos \left( {\pi + \theta } \right) = - \cos \theta $
 $
   = - 4\cos \left( {\dfrac{\pi }{3}} \right) \\
   = - 4\left( {\dfrac{1}{2}} \right) \\
   = - 2 \;
  $ using trigonometric value of $ \cos \left( {\dfrac{\pi }{3}} \right) = \dfrac{1}{2} $


 $
  y = 4\sin \left( {\dfrac{{4\pi }}{3}} \right) \\
   = 4\sin \left( {\pi + \dfrac{\pi }{3}} \right) \;
  $
Using Allied angle in trigonometry $ \sin \left( {\pi + \theta } \right) = - \sin \theta $
 $
   = - 4\sin \left( {\dfrac{\pi }{3}} \right) \\
   = - 4\left( {\dfrac{{\sqrt 3 }}{2}} \right) \\
   = - 2\sqrt 3 \;
  $ using trigonometric value of $ \sin \left( {\dfrac{\pi }{3}} \right) = \dfrac{{\sqrt 3 }}{2} $
Therefore, polar coordinates $ \left( {4,\dfrac{{4\pi }}{3}} \right) $ in rectangular coordinates are $ \left( { - 2, - 2\sqrt 3 } \right) $ .
So, the correct answer is “ $ \left( { - 2, - 2\sqrt 3 } \right) $”.

Note: A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length.