Question

The cartesian coordinates of the points, whose polar coordinates are $\left( 5,-\dfrac{\pi }{4} \right)$ ?
(a) $\left( \dfrac{5}{\sqrt{2}},\dfrac{5}{\sqrt{2}} \right)$
(b) $\left( \dfrac{5}{\sqrt{2}},\dfrac{-5}{\sqrt{2}} \right)$
(c) $\left( \dfrac{-5}{\sqrt{2}},\dfrac{5}{\sqrt{2}} \right)$
(d) $\left( \dfrac{-5}{\sqrt{2}},\dfrac{-5}{\sqrt{2}} \right)$

Answer 1

Hint: First, we should know how to convert polar coordinates to find cartesian coordinates points given in form of $\left( r,\theta \right)$ i.e. $\left( 5,-\dfrac{\pi }{4} \right)$ . Then we will use the formula $\left( x,y \right)=\left( r\cos \theta ,r\sin \theta \right)$ to find the cartesian coordinate points. Also, we will use $\cos \left( -\theta \right)=\cos \theta$ and $\sin \left( -\theta \right)=-\sin \theta$ when needed.

Complete step-by-step answer:
Here, we are given polar coordinates points i.e. $\left( 5,-\dfrac{\pi }{4} \right)$ which is equal to $\left( r,\theta \right)$ . So, from this we have to convert to cartesian coordinates points written as $\left( x,y \right)$ .
So, to find cartesian points we have a formula which we will use here given as $\left( x,y \right)=\left( r\cos \theta ,r\sin \theta \right)$ . Here, $r=5,\theta =-\dfrac{\pi }{4}$
So, using the above formula we get
$x=r\cos \theta$
On substituting the values, we get
$x=5\cos \left( -\dfrac{\pi }{4} \right)$
Now, we know that $\cos \left( -\theta \right)=\cos \theta$ and value of $\cos \left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}}$ so, using this we get value of x as
$x=5\cos \left( -\dfrac{\pi }{4} \right)=\dfrac{5}{\sqrt{2}}$ ………………………………(1)
Similarly, using the above formula we get
$y=r\sin \theta$
On substituting the values, we get
$x=5\sin \left( -\dfrac{\pi }{4} \right)$
Now, we know that $\sin \left( -\theta \right)=-\sin \theta$ and value of $\sin \left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}}$ so, using this we get value of y as
$y=5\sin \left( -\dfrac{\pi }{4} \right)=\dfrac{-5}{\sqrt{2}}$ ………………………………(2)
Thus, we get value of cartesian coordinates points as $\left( x,y \right)=\left( \dfrac{5}{\sqrt{2}},\dfrac{-5}{\sqrt{2}} \right)$ .
Hence, option (b) is the correct answer.

Note: Be careful which converting polar coordinates points to cartesian points because sometimes mistake happens when angle $\theta$ is given with minus sign and in hurry students use that minus sign with function cosine although there is no impact of minus sign on cosine function i.e. $\cos \left( -\theta \right)=\cos \theta$ . So, with this minus sign whole answer will be changed and will be $\left( \dfrac{-5}{\sqrt{2}},\dfrac{-5}{\sqrt{2}} \right)$ which will be wrong. So, do not make these silly mistakes.