Calculate angles of elevation and depression

The angle of depression is the angle formed by a horizontal line from the observer's eye and the line of sight going downward to an object. When you look down from a height (like from a balcony or cliff), the angle your sight line makes with the horizontal is the angle of depression.

Important property: The angle of elevation and angle of depression are equal because they are alternate angles formed by a horizontal line intersecting two parallel lines (the sight lines).

When solving problems involving angles of elevation and depression, follow these steps:

• Draw a clear diagram showing the observer's position, the object, and the horizontal line
• Identify the right-angled triangle formed
• Label the known side (adjacent to the angle) and the unknown side (opposite to the angle)
• Use the appropriate trigonometric ratio (usually tangent) to set up an equation
• Solve for the unknown using: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

Problem: From the top of a tower, the angle of depression of a point on the ground 10 metres away from the base of the tower is 60°. How high is the tower?

Solution:

Let A be the top of the tower, B be the base of the tower, and C be the point on the ground. The horizontal line through A meets the vertical line BC at point D, where AD is horizontal and AB is vertical (the height of the tower).

From the triangle ABC:

• $\angle ACB = 60°$ (angle of depression)
• $BC = 10$ m (distance from base of tower to point C)

Using the tangent ratio:

$\tan 60° = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BC}$

$AB = BC \times \tan 60°$

$AB = 10 \times 1.732 = 17.32 \text{ m}$

Therefore, the height of the tower is 17.32 metres.

Problem: A man whose height is 172 cm notices that his shadow measures 158 cm in length. Find the angle of elevation of the sun.

Solution:

Let XZ represent the man's height (172 cm) and YZ represent the length of his shadow (158 cm). The angle of elevation of the sun is the angle $\theta$ at Z.

Using the tangent ratio:

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{XZ}{YZ} = \frac{172}{158} = 1.089$

$\theta = \tan^{-1}(1.089) = 47°26'$

Therefore, the angle of elevation of the sun is approximately 47°.

For angle of elevation or depression problems:

• $\tan\theta = \frac{\text{height}}{\text{distance}}$
• Height = distance × $\tan\theta$
• Distance = height ÷ $\tan\theta$

In Tanzania, construction workers use angles of elevation to determine the height of buildings, towers, or bridges. For example, when building a water tower in a village, a technician can measure the horizontal distance from the tower's base to an observation point, then measure the angle of elevation to the top using simple tools. By applying $\tan\theta = \frac{\text{height}}{\text{distance}}$, the technician can calculate the tower's height without needing to climb it, ensuring proper design and safety standards.