**If the angle of elevation of the top of a vertical 25 m high building is 41.333o, how far is it to the building?**

To solve this problem, we can use trigonometry, specifically the tangent function, which relates the angle of elevation to the opposite side (the height of the building) and the adjacent side (the distance to the building).

The formula is:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Here:
- \(\theta\) is the angle of elevation, which is \(41.333^\circ\).
- The "opposite" side is the height of the building, which is \(25\) meters.
- The "adjacent" side is the distance to the building, which we will call \(d\).

So,

\[
\tan(41.333^\circ) = \frac{25}{d}
\]

To find \(d\), rearrange the formula:

\[
d = \frac{25}{\tan(41.333^\circ)}
\]

Now, calculate the value of \(d\):

\[
d \approx \frac{25}{\tan(41.333^\circ)} \approx \frac{25}{0.8761} \approx 28.54 \text{ meters}
\]

So, the distance to the building is approximately **28.54 meters**.

Would you like further details or have any questions?

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Here are 5 related questions to consider:
1. How would the distance change if the building height increased to 30 meters?
2. What if the angle of elevation was 30 degrees instead? How would that affect the distance?
3. Can you derive the formula for the distance using the sine or cosine functions instead?
4. How would you calculate the angle of elevation if you know the distance and height of the building?
5. How does the concept of angle of elevation apply in navigation or surveying?

**Tip:** Remember that the tangent function is useful for right-angle triangles when you know the opposite side and want to find the adjacent side, or vice versa.

Learn how to calculate the distance to a 25-meter high building using the angle of elevation of 41.333 degrees. Understand the application of trigonometric functions, specifically the tangent function, in solving real-world distance problems involving right triangles.

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