Question

How do you find the rectangular coordinates given $(6,150^\circ )$ ?

Answer 1

Hint : In this question, we are given an ordered pair, it is also known as the coordinates of a point. We see that the first one is an integer and the second one is an angle, so the given coordinates are of the form $(r,\theta )$ . There are two types of coordinates for plotting a point on the graph paper namely rectangular coordinate system and polar coordinate system. The given coordinates are the polar coordinates and we have to find the rectangular coordinates.
A right-angled triangle is formed by x, y and r, where x is the base, y is the height and r is the hypotenuse of the right-angled triangle, so by Pythagoras theorem, we get – ${x^2} + {y^2} = {r^2}$ and by trigonometry we get –
$\cos \theta = \dfrac{{base}}{{hypotenuse}} = \dfrac{x}{{\sqrt {{x^2} + {y^2}} }} = \dfrac{x}{r} \\\ \Rightarrow x = r\cos \theta \;$
And similarly $y = r\sin \theta$
Using this information, we can find the value of x and y.

Complete step by step solution:
We are given the polar coordinates are $(6,150^\circ )$ so we get $r = 6$ and $\theta = 150^\circ$
We know that
$x = r\cos \theta \\\ \Rightarrow x = 6\cos (150^\circ ) \\\ \Rightarrow x = 6\cos (180^\circ - 30^\circ ) \;$
We know $\cos (180 - x) = - \cos x$
$\Rightarrow x = 6( - \cos 30^\circ ) \\\ \Rightarrow x = - 6 \times \dfrac{{\sqrt 3 }}{2} \\\ \Rightarrow x = - 3\sqrt 3 \;$
$y = r\sin \theta \\\ \Rightarrow y = 6\sin (150^\circ ) \\\ \Rightarrow y = 6(\sin 180^\circ - 30^\circ ) \;$
We know that $\sin (180 - x) = \sin x$
$\Rightarrow y = 6\sin 30^\circ \\\ \Rightarrow y = 6 \times \dfrac{1}{2} \\\ \Rightarrow y = 3 \;$
Hence when the polar coordinates are $(6,150^\circ )$ , the rectangular coordinates are $( - 3\sqrt 3 ,3)$
So, the correct answer is “ $( - 3\sqrt 3 ,3)$ ”.

Note : 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. The rectangular coordinate system is of the form $(x,y)$ and is the most commonly used coordinate system, where x is the distance of this point from the y-axis and y is the distance of the point from the x-axis.