Question

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

Answer 1

We will have the following:

`r=\tan ^2(\theta)\csc (\theta)\Rightarrow r=((\sin(\theta))/(\cos(\theta)))^2((1)/(\sin(\theta)))`
`\Rightarrow r=(\sin^2(\theta))/(\sin(\theta)\cos^2(\theta))\Rightarrow r=(\sin(\theta))/(\cos^2(\theta))`

Now, we corroborate with one of the expression, in our case we will analyze y = x^2:

`r\sin (\theta)=(r\cos (\theta))^2\Rightarrow r\sin (\theta)=r^2\cos ^2(\theta)\Rightarrow r=(\sin (\theta))/(\cos ^2(\theta))`

So, the expression taht represents the value given is:

`y=x^2`