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To determine the height of the Eiffel Tower using the provided diagram and trigonometric values, we can follow these steps:

### Given:
- The observer is 100 meters away from the base of the Eiffel Tower.
- The angle of elevation from the observer to the top of the Eiffel Tower is \( 45^\circ \).
- There is also a smaller angle \( 30^\circ \) from a different point (likely closer to the base), but it is the 45° angle that is crucial here.

### Step 1: Apply Trigonometric Function
Since the angle of elevation is \( 45^\circ \) and we are given that:
\[ \sin 45^\circ = \frac{\sqrt{2}}{2} \]
We can use the tangent function, which relates the height of the tower to the distance from the observer:
\[ \tan 45^\circ = \frac{\text{Height of the Eiffel Tower}}{\text{Distance to the Tower}} \]

From trigonometric identities, we know:
\[ \tan 45^\circ = 1 \]

So:
\[ 1 = \frac{h}{100} \]

### Step 2: Solve for Height \( h \)
\[ h = 100 \times 1 = 100 \text{ meters} \]

Thus, the height of the Eiffel Tower based on this trigonometric calculation is **100 meters**.

Would you like more details or have any questions?

### Relative Questions:
1. How do you calculate the height if the angle was different?
2. Why is the tangent function used in this type of problem?
3. What if the distance was increased to 200 meters? How would that affect the height calculation?
4. How does changing the angle of elevation affect the calculation of height?
5. Can we calculate the height using \( \sin \) or \( \cos \) instead of \( \tan \)?

### Tip:
In problems involving right triangles and elevation angles, the tangent function is most often used because it directly relates the height of an object to the horizontal distance from the observer.

What is the height of the Eiffel Tower based on the given angles (30° and 45°) and the distance of 100m?

Learn how to calculate the height of the Eiffel Tower using trigonometry with angles of elevation and a 100m distance. This step-by-step solution involves trigonometric ratios, including the tangent function, to find the correct height based on provided values for sin(45°), sin(30°), and sin(15°).

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