Question

Find the rectangular coordinate of the point $\left( {5,300} \right).$

Answer 1

Hint:To convert Polar coordinates$\left( {r,\theta } \right)$to rectangular coordinates $\left( {x,y} \right)$,
we have the equation:
$
x = r\cos \theta \\
y = r\sin \theta \\
$
So by using the above equation and substituting the needed values we can find the rectangular coordinate corresponding to polar coordinate $\left( {5,300} \right).$

Complete step by step solution:
Given
$\left( {5,300} \right).....................\left( i \right)$
We know that rectangle coordinates are the Cartesian coordinates seen in the Cartesian plane which is represented by $\left( {x,y} \right)$and polar coordinates give the position of a point in a plane by using the length$r$and the angle made to the fixed point $\theta $, and is represented by $\left( {r,\theta }
\right).$
We know that (i) which is a polar coordinate is to be converted to a rectangular coordinate.
For that we can use the formula:
$
x = r\cos \theta .................\left( {ii} \right) \\
y = r\sin \theta ..................\left( {iii} \right) \\
$
So by substituting the values of $r\,\,{\text{and}}\,\,\theta $ in the equation (ii) and (iii) we can find the
values of $x\,\,{\text{and}}\,\,y.$
Now we know that on comparing (i) we can write:
$r = 5$
$\theta = 300 = \left( {2\pi - \dfrac{\pi }{3}} \right)$
i.e. changing $\theta $ from degrees to radians.
Now substituting the values of $r\,\,{\text{and}}\,\,\theta $ in the equation (ii) and (iii), we get:
$
\Rightarrow x = r\cos \theta = 5 \times \cos \left( {2\pi - \dfrac{\pi }{3}} \right) \\
\,\,\,\,\,\,\,\,\,x = 5 \times \dfrac{1}{2} \\
$
$ \Rightarrow x = \dfrac{5}{2}.......................\left( {iv} \right)$
Now for finding y:
$
\Rightarrow y = r\sin \theta \\
\,\,\,\,\,\,\,\,\,\,\, = 5 \times \sin \left( {2\pi - \dfrac{\pi }{3}} \right) \\
\,\,\,\,\,\,\,\,\,\,\, = 5 \times - \dfrac{{\sqrt 3 }}{2} \\
\Rightarrow y = - \dfrac{{5\sqrt 3 }}{2}..................\left( v \right) \\
$
So from (iv) and (v) we have got the values of $x = \dfrac{5}{2}\,\,{\text{and}}\,\,y = - \dfrac{{5\sqrt 3
}}{2}$, which are our rectangular coordinates.
Therefore the rectangular coordinate of the point $\left( {5,300} \right)$ is $\left( {\dfrac{5}{2}, -
\dfrac{{5\sqrt 3 }}{2}} \right)$.

Note: We know that to convert a polar coordinate$\left( {r,\theta } \right)$to a rectangular coordinate$\left(
{x,y} \right)$, we can use the formula:
$
x = r\cos \theta \\
y = r\sin \theta \\
$
In a similar manner convert rectangular coordinate$\left( {x,y} \right)$to a polar coordinate $\left( {r,\theta } \right)$, we can use the formula:
$
r = \sqrt {\left( {{x^2} + {y^2}} \right)} \\
\theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right) \\
$
Also while choosing $\theta $ it’s better to choose it in radians since when $\theta $ is in radians the calculations become much easier.