Question

How can you convert $r = \sin \theta$ to the rectangular equation?

Answer 1

Hint : In order to solve this question we will have to first let the terms in some other way then we will go for the appropriate manner to find the general equation and after it we will have to find the equation in the rectangular form.

Complete step-by-step answer :
For solving this question, first we will have to. When converting between polar coordinates and rectangular coordinates it is much straightforward to convert from polar coordinates to rectangular coordinates. However the conversion from rectangular coordinates to polar coordinates requires more work. When converting equations, it is more complicated to convert from polar to rectangular form.
To change a polar equation to a rectangular equation we need to place the following values:
$x = r\cos \theta \\\ y = r\sin \theta \\\ {r^2} = {x^2} + {y^2} \;$
We have
$r = \sin \theta$
Multiplying both sides by $r$ ,
${r^2} = r\sin \theta$
From the equations above, we obtain

$${x^2} + {y^2} = y \\\ {x^2} + {y^2} - y = 0 \;$$

Thus, we get
${x^2} + {y^2} - y = 0$
Which is the rectangular equation.
So, the correct answer is “ ${x^2} + {y^2} - y = 0$”.

Note : Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems. A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional space by a set of numbers. In rectangular coordinates these numbers are interpreted, roughly speaking, as the lengths of the sides of a rectangle.