Question

How do you convert $ r = 1 - \cos \theta $ into cartesian form?

Answer 1

Hint: First we will evaluate the right-hand of the equation and then further the left-hand side of the equation. We will use the following equations to convert the polar coordinates to cartesian coordinates:
  $
  x = r\cos \theta \\
  y = r\sin \theta \\
  r = \sqrt {{x^2} + {y^2}} \;
  $

Complete step-by-step answer:
We will start off by solving the right-hand side of the equation. Here, we will be using the equations
 $
  x = r\cos \theta \\
  y = r\sin \theta \\
  r = \sqrt {{x^2} + {y^2}} \;
  $ to convert the polar coordinates to cartesian coordinates.
Here, our equation is $ r = 1 - \cos \theta $ .
We can write the equation as $ \cos \theta = 1 - r $ .
From the above mentioned equation, we can write,
 $ \dfrac{x}{r} = 1 - r $
Now if we cross multiply the terms, the equation becomes,
 $ x = r - {r^2} $
Now, substitute the value of $ r $ in the equation.
 $
  x = \sqrt {{x^2} + {y^2}} - \left( {{x^2} + {y^2}} \right) \\
  x + \sqrt {{x^2} + {y^2}} = \left( {{x^2} + {y^2}} \right) \;
  $
Hence, the equation in cartesian form will be $ r = 1 - \cos \theta $ .
So, the correct answer is “$ x + \sqrt {{x^2} + {y^2}} = \left( {{x^2} + {y^2}} \right)$”.

Note: Converting between polar and Cartesian coordinate systems is relatively simple. Just take the cosine of $ \theta $ to find the corresponding Cartesian x coordinate, and the sine of $ \theta $ to find y coordinate. Basic trigonometry makes it easy to determine polar coordinates from a given pair of Cartesian coordinates. When we know a point in Cartesian Coordinates $ \,(x,y) $ and we want it in Polar Coordinates $ (r,\theta ) $ we solve a right triangle with two known sides.