To determine the height of an object using trigonometric ratios, you will measure angles and distances and then apply relevant trigonometric functions.

To calculate the height of an object effectively, you can utilize the fundamental trigonometric ratios: sine, cosine, and tangent. These ratios establish a relationship between the angles and the sides of a right-angled triangle. The tangent ratio is the most commonly used for this purpose, as it relates the height of the object (the side opposite the angle) to the distance from the object (the side adjacent to the angle).

Here is a clear step-by-step procedure:

1. Measure the Distance: Position yourself at a known distance from the base of the object. This distance will represent the adjacent side of your right triangle.

2. Measure the Angle: Employ a clinometer or a protractor to measure the angle of elevation from your eye level to the top of the object. This angle is defined as the angle formed between your line of sight to the top of the object and the horizontal ground.

3. Apply the Tangent Ratio: The tangent of the angle of elevation, denoted as $\theta$, is equal to the height of the object (the opposite side, $h$) divided by the distance from the object (the adjacent side, $d$). The relationship can be expressed with the following formula:

   $$\tan(\theta) = \frac{h}{d}$$

4. Solve for Height: Rearranging the equation allows you to isolate the height ($h$):

   $$h = d \times \tan(\theta)$$

Example: Suppose you are standing $10$ metres away from a tree, and you measure the angle of elevation to the top of the tree as $30^\circ$. You would calculate the height of the tree using the formula:

Using a calculator, you find that $\tan(30^\circ) \approx 0.577$. Thus, the height of the tree is calculated as follows:

This method allows you to accurately determine the height of any object, provided that you can measure both the distance to it and the angle of elevation.