What is the formula for the angle of elevation?

The formula for the angle of elevation is: \(\theta = \tan^{-1} \left(\frac{\text{opposite}}{\text{adjacent}}\right)\).

The angle of elevation is the angle formed between the horizontal ground and the line of sight when looking up at an object. To find this angle, you need to know the height of the object (opposite side) and the horizontal distance from the observer to the object (adjacent side). The formula uses the inverse tangent function, often written as \(\tan^{-1}\) or \(\arctan\), to calculate the angle.

For example, if you are standing 10 metres away from a tree and the tree is 5 metres tall, you can use the formula to find the angle of elevation. Here, the height of the tree is the opposite side, and the distance from the tree is the adjacent side. Plugging these values into the formula gives \(\theta = \tan^{-1} \left(\frac{5}{10}\right)\). Simplifying inside the brackets, you get \(\theta = \tan^{-1} (0.5)\). Using a calculator, you find that \(\theta\) is approximately 26.57 degrees.

This concept is particularly useful in trigonometry and helps in solving problems involving right-angled triangles. Understanding how to use the angle of elevation formula can also be applied in real-life situations, such as determining the height of a building or a mountain when you can measure the distance to it.