Question

# Rewrite the polar equation r=5\sin (θ)r=5\sin (θ) as a Cartesian equation.

Equations
Rewrite the polar equation $$r=5\sin (θ)r=5\sin (θ)$$ as a Cartesian equation.

2021-06-09
Step 1
To express the given curve in cartesian coordinates (x and y), using the transformation equations from the polar to the cartesian
Step 2
Transformation equations:
$$x=r\cos 0, y=r\sin 0, r^{2}=x^{2}+y^{2}$$...(1)
Given
$$r=5\sin 0$$...(2)
Step 3
Mutiply (2) by r and use the relation $$r^{2}=x^{2}+y^{2}$$ to obtain the required Cartesian equation
$$r\times (2):r^{2}=5r\sin 0$$
Apply (1) to get
$$x^{2}+y^{2}=5y$$, or
$$x^{2}+y^{2}-5y=0$$
Step 4
Answer: $$x^{2}+y^{2}-5y=0$$