The tangent function, $\tan(\theta)$, is undefined whenever the cosine of the angle equals zero. This occurs at specific angles, where the denominator in the ratio $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ becomes zero.

For the given options:

• $\tan(90^\circ)$ and $\tan(270^\circ)$ are undefined because the cosine of these angles is zero.
• Therefore, the correct answer is "at 90 and 270 degrees".

Let me know if you want further details on why the tangent function is undefined at these angles!

Here are five related questions to explore further:

1. What happens to the tangent function at 0°, 180°, and 360°?
2. How can you determine the undefined points of any trigonometric function?
3. What is the relationship between the tangent and sine/cosine functions?
4. How do you find the period of the tangent function?
5. What are the asymptotes of the tangent function?

Tip: The tangent function has vertical asymptotes wherever the cosine function equals zero.