How do you use the angle of depression to find a distance?

You use the angle of depression to find a distance by applying trigonometric ratios, specifically the tangent function.

When you are given the angle of depression, it is the angle formed by the line of sight from an observer looking downwards to an object and the horizontal line from the observer's eye level. To find the distance, you need to know either the height from which the observer is looking or the horizontal distance to the object.

Let's break it down with an example. Imagine you are standing on a cliff and looking down at a boat on the sea. The angle of depression from your eyes to the boat is 30 degrees, and you know the height of the cliff is 50 metres. To find the horizontal distance from the base of the cliff to the boat, you can use the tangent function in trigonometry, which relates the angle of a right triangle to the ratio of the opposite side (height of the cliff) to the adjacent side (horizontal distance).

The formula is:
\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \]

In this case:
\[ \tan(30^\circ) = \frac{50}{\text{distance}} \]

Rearranging to solve for the distance:
\[ \text{distance} = \frac{50}{\tan(30^\circ)} \]

Using a calculator, you find:
\[ \tan(30^\circ) \approx 0.577 \]

So:
\[ \text{distance} = \frac{50}{0.577} \approx 86.6 \text{ metres} \]

Therefore, the horizontal distance from the base of the cliff to the boat is approximately 86.6 metres. This method can be applied to any situation where you know the angle of depression and one side of the right triangle, allowing you to find the unknown distance.