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

• A. $y=0$
• B. $x=0$
• C. $x^2+y^2=0$
• D. $r=0$

Answer: (C)

Verified step by step guidance

1. Recall the given polar equation: \(r = (\tan 2\pi)(\csc \pi)\).
2. Evaluate the trigonometric functions \(\tan 2\pi\) and \(\csc \pi\) using their known values on the unit circle: \(\tan 2\pi = 0\) and \(\csc \pi = \frac{1}{\sin \pi}\), noting that \(\sin \pi = 0\) which makes \(\csc \pi\) undefined.
3. Since \(\csc \pi\) is undefined, consider the implications for the product \((\tan 2\pi)(\csc \pi)\) and how it affects the value of \(r\) in the polar equation.
4. Recognize that if \(r\) equals zero or is undefined, the point lies at the origin or is not defined, which corresponds to the rectangular equation \(x^2 + y^2 = 0\) representing the origin.
5. Conclude that the rectangular form of the given polar equation is \(x^2 + y^2 = 0\), which describes the point at the origin.