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

We can compare the polar coordinates to $\left( {r,\theta } \right)$ and find the values of r and $\theta$ . Then we can find the x coordinate, using the equation $x = r\cos \theta$ and y coordinate using the equation $y = r\sin \theta$ . Then we can write x and y as ordered pairs to get the required cartesian coordinates.

Complete step by step answer:

We know that the cartesian coordinates $\left( {x,y} \right)$ of a point in the polar form $\left( {r,\theta } \right)$ is given by the equations,
$x = r\cos \theta$ …. (1)
$y = r\sin \theta$ …. (2)
We have the polar coordinate given as $\left( { - 5, - \dfrac{\pi }{4}} \right)$ . On comparing with $\left( {r,\theta } \right)$ , we can say that,
$r = - 5$ and $\theta = - \dfrac{\pi }{4}$ .
Now we can substitute these values in equation (1) to find the x coordinate. So, we get,
$\Rightarrow x = - 5 \times \cos \left( { - \dfrac{\pi }{4}} \right)$
We know that $\cos \left( { - x} \right) = \cos x$ . So, we get,
$\Rightarrow x = - 5 \times \cos \left( {\dfrac{\pi }{4}} \right)$
We know that $\cos \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}$ . On substituting in the above equation, we get,
$\Rightarrow x = \dfrac{{ - 5}}{{\sqrt 2 }}$
So, the x coordinate is $\dfrac{{ - 5}}{{\sqrt 2 }}$
Now we can substitute these values in equation (2) to find the y coordinate. So, we get,
$\Rightarrow y = - 5 \times \sin \left( { - \dfrac{\pi }{4}} \right)$
We know that $\sin \left( { - x} \right) = - \sin x$ . So, we get,
$\Rightarrow y = 5 \times \sin \left( {\dfrac{\pi }{4}} \right)$
We know that $\sin \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}$ . On substituting in the above equation, we get,
$\Rightarrow y = \dfrac{5}{{\sqrt 2 }}$
So, the y coordinate is $\dfrac{5}{{\sqrt 2 }}$
Therefore, the cartesian coordinate of the given point is $\left( {\dfrac{{ - 5}}{{\sqrt 2 }},\dfrac{5}{{\sqrt 2 }}} \right)$
So, the correct answer is option C.

Note: In a Cartesian coordinate system, a point is located using its perpendicular distance from the X and Y axes. In a polar coordinate, a point is represented using the distance from the origin and the angle this shortest distance makes with the positive X axis. To convert a Cartesian coordinate to polar form, we use the following equations.
$r = \sqrt {{x^2} + {y^2}}$
$\theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)$