Question

What is the Cartesian form of $\left( {0,\pi } \right)$ ?

Answer 1

Hint : The given form is in polar form. The general polar form is $\left( {r,\theta } \right)$ . Hence, $r$ will be equal to 0 and $\theta$ will be equal to $\pi$ . Now, to convert polar form into Cartesian form, we will be using the formula
$x = r \times \cos \theta$ and $y = r \times \sin \theta$ . Using these formulas, we will get the Cartesian coordinates of the given polar coordinates.

Complete step by step solution:
In this question, we have to find the Cartesian form of $\left( {0,\pi } \right)$ .
The given form is in polar form.
First of all, what are Cartesian forms and polar forms?
Cartesian form:
Cartesian coordinates are used to mark how far along and how far up a point is.
Cartesian coordinates are represented by $\left( {x,y} \right)$ .
Polar form:
Polar coordinates are used to mark how far away and at what angle a point is.
Polar coordinates are represented by $\left( {r,\theta } \right)$ , where $r$ is the distance and $\theta$ is the angle.
Conversion of polar coordinates $\left( {r,\theta } \right)$ to Cartesian coordinates $\left( {x,y} \right)$ :
$\to x = r \times \cos \theta$ .
$\to y = r \times \sin \theta$ .
In our question, polar coordinates are $\left( {0,\pi } \right)$ . Therefore,
$r = 0$ and $\theta = \pi$ .
Therefore, Cartesian coordinates will be
$\to x = 0 \times \cos \pi \\\ \to x = 0 \;$
And,
$\to y = 0 \times \sin \pi \\\ \to y = 0 \;$
Therefore, $\left( {0,0} \right)$ will be our Cartesian form.
Hence, we have converted $\left( {0,\pi } \right)$ polar form into $\left( {0,0} \right)$ Cartesian form.
So, the correct answer is “{0,0}”.

Note : Conversion of Cartesian coordinates $\left( {x,y} \right)$ to polar coordinates $\left( {r,\theta } \right)$ :
Cartesian coordinates can be converted to polar coordinates using the formula
$\to r = \sqrt {{x^2} + {y^2}}$
$\to \theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)$
In our question, we have $x = 0,y = 0$
Therefore, $r = \sqrt {0 + 0} = 0$ and $\theta = {\tan ^{ - 1}}\left( {\dfrac{0}{0}} \right) = {\tan ^{ - 1}}0 = \pi$ .
Hence, we have converted the Cartesian form $\left( {0,0} \right)$ into polar form $\left( {0,\pi } \right)$ .