Question

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

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asked 2021-06-08
Rewrite the polar equation \(r=5\sin (θ)r=5\sin (θ)\) as a Cartesian equation.

Answers (1)

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: \(x^{2}+y^{2}-5y=0\)
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