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In the problem, we are asked to relate the angle \( \theta \) that the sun makes with a gnomon to the length of the shadow \( l \) and the height of the gnomon \( h \). Assuming \( \theta \) is the angle between the gnomon and the sun's rays, the tangent of this angle \( \tan(\theta) \) is given by the ratio of the opposite side to the adjacent side in a right triangle formed by the height of the gnomon and the length of the shadow on the ground. 

Therefore, the trigonometric equation relating \( \theta \), \( l \), and \( h \) is:
\[
\tan(\theta) = \frac{h}{l}
\]
This equation can be used to find any of the three variables (\( \theta \), \( l \), \( h \)) if the other two are known. 

Would you like more details or help with another part of this topic? Here are some follow-up questions you might consider:

1. How would you calculate the height of the gnomon if the angle and shadow length are given?
2. What happens to the length of the shadow as the angle \( \theta \) increases?
3. Can you solve for \( \theta \) if \( h \) and \( l \) are provided?
4. How does the position of the sun affect \( \theta \) throughout the day?
5. Is it possible to determine the time of day based on the angle \( \theta \) and the gnomon’s properties?

**Tip:** When using the tangent function in calculations, ensure your calculator is set to the correct mode (degrees or radians) based on the unit of the angle provided.

Suppose that Eratosthenes had modern trigonometry when he measured the angle at which the sun's rays strike the gnomon at Alexandria. If this were available, he could have measured the length of the shadow (l) on the ground, and the height (h) of the gnomon. Write a trigonometric equation that relates theta (θ) — the angle that the sun makes with the gnomon — to the length of the shadow and height of the gnomon.

Learn how to write a trigonometric equation relating the angle formed by the sun with a gnomon to the length of the shadow and height of the gnomon. This problem uses the tangent function in trigonometry, involving key concepts of right triangles and angle measurement. Suitable for high school students.

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