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.