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) $ .