Solving Height and Distance Problems: Understanding Angle of Depression

This problem involves calculating the height of a tower using the concept of the angle of depression. Let's break down the scenario and the steps to find the solution.

We have a tower and a building. The distance between the base (foot) of the building and the base of the tower is given as 50 m.

The angle of depression is the angle measured downwards from a horizontal line when an observer looks at an object below the horizontal. In this case, the observer is at the top of the tower, and the object is the foot of the building.

The angle of depression from the top of the tower to the foot of the building is given as 60°. This angle is formed by the horizontal line at the top of the tower and the line of sight to the foot of the building.

Relating Angle of Depression to Angle of Elevation

An important concept in trigonometry problems involving heights and distances is the relationship between the angle of depression and the angle of elevation. The angle of elevation is the angle measured upwards from a horizontal line when an observer looks at an object above the horizontal.

When looking from the top of the tower to the foot of the building (angle of depression), the horizontal line at the top of the tower is parallel to the ground (which is the horizontal line at the foot of the building). The line of sight connecting the top of the tower and the foot of the building acts as a transversal line.

Therefore, the angle of depression from the top of the tower to the foot of the building is equal to the angle of elevation from the foot of the building to the top of the tower. This is because they are alternate interior angles formed by a transversal intersecting two parallel lines.

So, the angle of elevation from the foot of the building to the top of the tower is also 60°.

Setting up the Trigonometric Problem

We can visualize this situation as a right-angled triangle:

• The vertical side of the triangle is the height of the tower (which we need to find, let's call it 'h').
• The horizontal side of the triangle is the distance between the foot of the tower and the foot of the building (given as 50 m).
• The angle of elevation at the foot of the building is 60°.

In this right-angled triangle, with respect to the angle of elevation (60°):

• The height of the tower ('h') is the side opposite to the angle.
• The distance between the bases (50 m) is the side adjacent to the angle.

We can use the tangent trigonometric ratio, which relates the opposite side to the adjacent side in a right-angled triangle:

\(\tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}}\)

Calculating the Height of the Tower

Using the values from our problem:

• Angle \(\theta = 60^\circ\)
• Opposite side = height of the tower (h)
• Adjacent side = distance between bases (50 m)

So, we have:

\(\tan(60^\circ) = \frac{h}{50}\)

We know the standard value of \(\tan(60^\circ)\) is \(\sqrt{3}\).

Substituting this value into the equation:

\(\sqrt{3} = \frac{h}{50}\)

To find 'h', we multiply both sides of the equation by 50:

\(h = 50 \times \sqrt{3}\)

\(h = 50\sqrt{3}\)

So, the height of the tower is \(50\sqrt{3}\) meters.

This calculation shows that the height of the tower is 50 times the square root of 3 meters.

Revision Table: Key Trigonometric Ratios

Here is a quick look at the basic trigonometric ratios commonly used in such problems:

Standard values for common angles like 30°, 45°, and 60° are essential for solving these problems quickly.

Additional Information: Applications of Trigonometry

Trigonometry, specifically the concepts of angles of elevation and depression, is widely used in various real-world applications:

• Surveying: To determine the height of buildings, mountains, or the distance across rivers.
• Navigation: Used in air and sea navigation to calculate distances and directions.
• Engineering: In construction and architecture for designing structures and calculating forces.
• Astronomy: To calculate distances between celestial bodies.

Understanding how to use trigonometric ratios and the relationship between angles of elevation and depression is fundamental to solving many practical problems involving heights and distances.