Question

Which of the following represents the polar equation r = ( cot ( 2 θ ) ) ( csc ( θ ) ) as a rectangular equation?

Answer 1

Recall the following rules for identifying the rectangular coordinates (x,y) from the polar coordinates (r,θ):

`\begin{gathered} x=r\cos\theta \\ y=r\sin\theta \end{gathered}`
`r=√(x^2+y^2)`

First, rewrite cot(2θ)csc(θ) in terms of sines and cosines:

`\begin{gathered} r=(\cot2\theta)(\csc\theta) \\ \\ =(\cos2\theta)/(\sin2\theta)\cdot(1)/(\sin\theta) \\ \\ =(1-2\sin^2\theta)/(2\sin\theta\cos\theta)\cdot(1)/(\sin\theta) \\ \\ =(1-2\sin^2\theta)/(2\sin^2\theta\cos\theta) \end{gathered}`

Multiply both numerator and denominator by r^3 and order the factors in such a way that the expressions for x and y are made evident:

`\begin{gathered} r=(1-2\sin^2\theta)/(2\sin^2\theta\cos\theta) \\ \\ \Rightarrow r=(r^3(1-2\sin^2\theta))/(r^3(2\sin^2\theta\cos\theta)) \\ \\ =(r(r^2-2(r^2\sin^2\theta)))/(2(r^2\sin^2\theta)(r\cos\theta)) \\ \\ =(r(r^2-2(r\sin\theta)^2))/(2(r\sin\theta)^2(r\cos\theta)) \end{gathered}`

Simplify and replace the expressions for rsin(θ) and rcos(θ) as well as r:

`\begin{gathered} r=(r(r^2-2(r\sin\theta)^2))/(2(r\sin\theta)^2(r\cos\theta)) \\ \\ \Rightarrow1=((r^2-2(r\sin\theta)^2))/(2(r\sin\theta)^2(r\cos\theta)) \\ \\ \Rightarrow1=((x^2+y^2-2(y)^2))/(2(y)^2(x)) \\ \\ \Rightarrow1=((x^2+y^2-2y^2))/(2y^2x) \\ \\ \Rightarrow1=((x^2-y^2))/(2y^2x) \\ \\ \Rightarrow2y^2x=x^2-y^2 \end{gathered}`

Isolate y^2 from the equation:

`\begin{gathered} \Rightarrow2y^2x+y^2=x^2 \\ \\ \Rightarrow y^2(2x+1)=x^2 \\ \\ \Rightarrow y^2=(x^2)/(2x+1) \end{gathered}`

Therefore, the polar equation can be written as a rectangular equation as follows:

`y^2=(x^2)/(2x+1)`