Question

rewrite the polar equation $r = 4\sin(\theta)$ as a cartesian equation.

Explanation:

Step1: Multiply both sides by \(r\)

$$r\times r = 4r\sin(\theta)$$
Since \(x = r\cos(\theta)\), \(y = r\sin(\theta)\) and \(r^{2}=x^{2}+y^{2}\), the equation becomes \(r^{2}=4y\).

Step2: Substitute \(r^{2}\)

Substitute \(r^{2}=x^{2}+y^{2}\) into \(r^{2}=4y\), we get \(x^{2}+y^{2}=4y\).

Step3: Complete the square for \(y\) terms

Rearrange the equation: \(x^{2}+y^{2}-4y = 0\).
For the \(y\) terms \(y^{2}-4y\), using the formula \((a - b)^2=a^{2}-2ab + b^{2}\), where \(a = y\) and \(2b=4\) (so \(b = 2\)), \(y^{2}-4y=(y - 2)^{2}-4\).
The equation \(x^{2}+y^{2}-4y = 0\) can be rewritten as \(x^{2}+(y - 2)^{2}-4=0\), or \(x^{2}+(y - 2)^{2}=4\).

Answer:

\(x^{2}+(y - 2)^{2}=4\)