https://r2.erweima.ai/mathbot/image/1726614602928p7efl1do.jpg

Let's analyze the problem from the image. It describes a scenario involving a lighthouse and a boat, where we are tasked with calculating the horizontal distance from the lighthouse to the boat. Here's a breakdown:

### Given:
- The **height of the lighthouse** is 8 meters.
- The **angle of depression** from the top of the lighthouse to the boat is 8°.
- The goal is to find the horizontal distance between the **foot of the lighthouse** and the boat.

This forms a right triangle where:
- The height of the lighthouse (8 meters) is the opposite side.
- The horizontal distance to the boat is the adjacent side.
- The angle of depression (8°) is given.

Using **trigonometry**, we can use the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Where:
- \(\theta = 8^\circ\)
- \(\text{opposite} = 8 \, \text{m}\)

Rearranging to solve for the adjacent side (the distance to the boat):
\[
\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)} = \frac{8}{\tan(8^\circ)}
\]

Let's calculate the value.The horizontal distance between the boat and the foot of the lighthouse is approximately **56.92 meters**.

Would you like more details on the steps? Here are five related questions to expand on this topic:

1. How would the distance change if the angle of depression were increased?
2. What if the height of the lighthouse were doubled? How would that affect the result?
3. How do trigonometric functions apply in real-world navigation problems?
4. Can this method be used to estimate distances in different situations, like from a mountain?
5. How does the tangent function behave for small vs. large angles?

### Tip:
When working with angles of depression or elevation, always visualize the right triangle and determine which sides (opposite, adjacent) are involved. This simplifies which trigonometric function to apply.

Desde la cima de un faro de 8 metros de altura se divisa una lancha con un ángulo de depresión de 8°. Calcula la distancia entre la lancha y el pie del faro en ese mismo instante.

Learn how to calculate the distance between a boat and a lighthouse using an 8° angle of depression and the height of the lighthouse. This problem uses trigonometric concepts like the tangent function to solve a right triangle problem.

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