The value of \( \tan 90^\circ \) (or \( \tan \frac{\pi}{2} \) radians) is **undefined**.

This is because the tangent function is defined as the ratio of the sine to the cosine:

\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]

At \( \theta = 90^\circ \), \( \sin 90^\circ = 1 \) and \( \cos 90^\circ = 0 \). Since division by zero is undefined, \( \tan 90^\circ \) does not have a defined value.

Would you like more details on the behavior of the tangent function near 90 degrees?

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Here are 5 related questions to expand on this concept:
1. What happens to the value of \( \tan \theta \) as \( \theta \) approaches 90°?
2. How does the tangent function behave for angles beyond 90°?
3. What is the value of \( \tan 0^\circ \)?
4. Can you explain the relationship between the sine, cosine, and tangent functions?
5. How can you graphically interpret \( \tan \theta \)?

**Tip:** The tangent function has vertical asymptotes at odd multiples of 90°, indicating points where the function becomes undefined.

Explore why the value of tan 90° is undefined with a detailed explanation of the tangent function as the ratio of sine and cosine. This trigonometric problem highlights key concepts in trigonometry, including vertical asymptotes and the behavior of functions at critical points. Suitable for students in Grades 10-12.

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