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.