Angle of Elevation and Depression

Trigonometry is a fascinating branch of mathematics that studies the relationships between the sides and angles of triangles. A very practical application of trigonometry is in the area of measuring heights and distances, especially when it deals with elevation and depression angles. This subject not only helps us solve real-life problems but also enhances our understanding of spatial relationships.

What is angle of elevation?

Imagine you are standing on the ground and looking at an object, such as the top of a tree, tower, or mountain. The angle of elevation is the angle between the horizontal line from the observer's eye to the object. It is the angle that someone would need to "raise" their eyes to see the top of the object from where they are standing.

    Consider the following scenario:


               A (top of the tree)
               ,
              ,
     H / |
            ,
           ,
        θ /_____|
         /e (eye level or horizontal line)
        B
    

in this instance:

• A is the top of the tree.
• B is the point where you are standing.
• E is your eye level or the horizontal line from your observation point.

The line AB represents the height (h) you want to measure. The angle θ between the line BE (your line of sight) and the horizontal line is the angle of elevation.

What is angle of depression?

Now, let us discuss the angle of depression. Suppose you are standing at the top of a lighthouse and looking down at a boat in the sea. The angle of depression is the angle between the horizontal line from the eye of the observer to the object below.

    Consider the following scenario:


        A (eye level of the person at the top of the lighthouse)
        ,
           ,
            ,
             ,
              | θ
               _____B (boat)
                D (horizontal line from observer's eye level)
    

In this illustration:

• A is the eye level of the person at the top of the lighthouse.
• B There is a boat on the sea.
• D is a horizontal line drawn parallel to the line coming from the person's eyes.

The angle θ is called the angle of depression. It is important to know that when two parallel lines are crossed by a transversal, the angle of elevation from the point of view of the observer is equal to the angle of depression from the line of sight due to the alternate internal angles.

Understanding through simple mathematical formulas

To calculate the height of an object using the angles of elevation and depression, we often use trigonometric ratios. The most common trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). In problems involving height and distance, tan θ ratio is used most often because it relates the opposite side (height) to the adjacent side (distance) in a right triangle:

    Tangent of angle = opposite side / adjacent side
       tan θ = h/d
    

Where:

• θ is the angle of elevation or depression.
• h is the height (or depth in case of depression) of the object.
• d is the distance from the observer to the base of the object.

Example 1: Angle of elevation

Let's solve a practical problem using the angle of elevation:

Question: A person standing 50 m away from a tree sees the top of the tree at an angle of elevation of 30°. What is the height of the tree?

Solution:

1. Identify the given values: distance (d) = 50m, angle of elevation (θ) = 30°.
2. Use the formula for the tangent of the angle of elevation:

    tan θ = opposite side / adjacent side

    tan 30° = h / 50

3. Since tan 30° = 1/√3, substituting gives:

    1/√3 = h / 50

4. Solve for h (the height of the tree):

    h = 50 / √3 ≈ 28.87 m

The height of the tree is approximately 28.87 metres.

Example 2: Angle of depression

Now, let's look at a problem involving the angle of depression:

Problem: A person at a height of 60 m in a lighthouse sees a boat at a horizontal distance of 80 m from the base of the lighthouse. What is the angle of depression towards the boat?

Solution:

1. Identify the given values: height (h) = 60m, distance (d) = 80m.
2. Use the formula for the tangent of the angle of depression:

    tan θ = h / d

    tan θ = 60 / 80

3. Simplification:

    tan θ = 0.75

4. Find θ using the arctan function:

    θ = arctan(0.75) ≈ 36.87°

The angle of depression is about 36.87°.

Why do angles of elevation and depression matter?

The concepts of elevation and depression angles are essential in the real world because they help us measure unattainable distances and heights. Architects, engineers, aviators, and sailors often use these calculations to solve practical problems.

For example, when a tall building is built, its height is often measured from a distance using the angle of elevation. Similarly, navigators use the angle of depression to find the distance of objects observed from a vantage point, such as a lighthouse.

Additional exercise examples

To understand more, let's solve a few more examples:

Example 3: Angle of elevation

Situation: A ladder rests against a wall such that its top reaches a window 7 m above the ground, and the angle of elevation is 45°. How far is the base of the ladder from the wall?

 given:
      Opposite side (height) = 7 m,
      Angle of elevation (θ) = 45°

We use the formula:

 tan θ = opposite/adjacent
        Tan 45° = 7 / day

Since tan 45° = 1, we get:

 1 = 7 / day
        d = 7

The base of the ladder is at a distance of 7 m from the wall.

Example 4: Angle of depression

Situation: From the top of a building 50 m high, the angle of depression of a car parked next to the building is 60°. How far is the car from the base of the building?

 given:
       Height of the building (h) = 50 m,
       Angle of depression (θ) = 60°

We use the formula:

 tan θ = h / d
        Tan 60° = 50 / day

Since tan 60° = √3, we calculate:

 √3 = 50 / day
        d = 50 / √3 ≈ 28.87 m

The car is approximately 28.87 metres away from the base of the building.

Conclusion

Angles of elevation and depression are crucial in understanding how we can calculate height and distance using trigonometry. By mastering these concepts, you open yourself up to efficiently solving many real-world geometry problems. Keep practicing with different scenarios to thoroughly understand how these angles work, and you'll find it much easier to deal with various trigonometric applications in your daily life or future career paths.

Comments