How do you solve problems involving angles of elevation and depression?

To solve problems involving angles of elevation and depression, use trigonometric ratios and right-angled triangle properties.

Angles of elevation and depression are commonly found in real-world scenarios, such as measuring the height of a building or the distance to a ship at sea. The angle of elevation is the angle between the horizontal line of sight and the line of sight up to an object. Conversely, the angle of depression is the angle between the horizontal line of sight and the line of sight down to an object.

To solve these problems, you typically use trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios relate the angles of a right-angled triangle to the lengths of its sides. For example, if you know the angle of elevation and the distance from the observer to the base of the object, you can use the tangent ratio (tan = opposite/adjacent) to find the height of the object.

Here’s a step-by-step approach:
1. **Draw a Diagram**: Sketch the scenario to visualise the right-angled triangle.
2. **Identify Known Values**: Determine which sides and angles you know.
3. **Choose the Right Trigonometric Ratio**: Depending on the known values, use sin (opposite/hypotenuse), cos (adjacent/hypotenuse), or tan (opposite/adjacent).
4. **Set Up the Equation**: Write the trigonometric equation based on the chosen ratio.
5. **Solve for the Unknown**: Rearrange the equation to solve for the unknown side or angle.
6. **Check Units and Reasonableness**: Ensure your answer makes sense in the context of the problem.

For example, if you need to find the height of a tree and you know the distance from the tree and the angle of elevation, you can use the tangent function. If the distance is 30 metres and the angle of elevation is 45 degrees, the height (h) can be found using tan(45) = h/30. Since tan(45) = 1, h = 30 metres.