Question

Convert the equation to polar form.
1. x^(2) - y^(2) = 1

Convert the polar equation to rectangular coordinates.
1. r = 2csc θ

2. r = 3(1-sinθ)

3. r = 2 - cosθ

4. r = 4/(1+2sinθ)

5. r^2 = sin2θ

Answer 1

@johnweldon1993

Answer 2

@Preetha

Answer 3

Using the standard conversions, you have \(x=r\cos t\) and \(y=r\sin t\), so the equation becomes
\[r^2\cos^2t-r^2\sin^2t=1\]
For the others, you can use the same tactic, plus the fact that \(x^2+y^2=r^2\) (and so \(r=\sqrt{x^2+y^2}\)):
\[r=2\csc t\implies r\sin t=2\implies\cdots\]\[r=3(1-\sin t)=3-3\sin t\implies r^2=3r-3r\sin t\implies \cdots\]\[r=2-\cos t\implies r^2=2r-r\cos t\implies\cdots\]\[r=\frac{4}{1+2\sin t}\implies r+2r\sin t=4\implies\cdots\]\[r^2=\sin2t=2\sin t\cos t\implies r^4=2(r\sin t)(r\cos t)\implies\cdots\]