Applications of Trigonometry: Heights and Distances

TL;DR

You'll learn to use trigonometry to find unknown heights or distances by forming right-angled triangles from real-world scenarios. We'll focus on angles of elevation and depression, which are crucial for setting up these problems correctly. Mastering these concepts will help you solve problems involving towers, buildings, and observers.

1. The Mental Model

Imagine you're an architect or a surveyor. You can't always directly measure the height of a tall building or the distance across a river. Instead, you use angles and known distances to "calculate" those measurements indirectly, forming an invisible right-angled triangle in your mind.

2. The Core Material

When we talk about "Heights and Distances," we're essentially applying the basic trigonometric ratios (sine, cosine, tangent) to practical problems. The key is to visualize and draw a right-angled triangle from the given situation.

Angles of Elevation and Depression

These are super important! They tell you how to set up your triangle.

• Angle of Elevation: This is the angle formed when you look up from a horizontal line to an object above you. Think of looking up at the top of a tree.
• Angle of Depression: This is the angle formed when you look down from a horizontal line to an object below you. Think of looking down from a cliff at a boat in the water.

Remember: The angle of elevation from point A to point B is equal to the angle of depression from point B to point A, assuming they are at the same horizontal level relative to each other. This is because they are alternate interior angles when a transversal cuts two parallel lines (the horizontal lines).

graph TD
    A["Observer's Eye (Higher Point)"] -->|Horizontal Line| H["(Reference)"]
    H -->|Line of Sight (Downward)| D["Object (Lower Point)"]
    subgraph Angle of Depression
        H --- Depression_Angle_Arc
        Depression_Angle_Arc --> D
    end

    B["Observer's Eye (Lower Point)"] -->|Horizontal Line| I["(Reference)"]
    I -->|Line of Sight (Upward)| E["Object (Higher Point)"]
    subgraph Angle of Elevation
        I --- Elevation_Angle_Arc
        Elevation_Angle_Arc --> E
    end

    style H fill:#fff,stroke:#333,stroke-width:2px,stroke-dasharray: 5 5
    style I fill:#fff,stroke:#333,stroke-width:2px,stroke-dasharray: 5 5
    linkStyle 1 stroke-dasharray: 5 5
    linkStyle 4 stroke-dasharray: 5 5

Steps to Solve Problems

1. Draw a Diagram: This is crucial! Always draw a neat, labelled diagram representing the situation. Mark the knowns (distances, angles) and the unknown you need to find.
2. Identify the Right Triangle: Your diagram should clearly show at least one right-angled triangle.
3. Choose the Correct Trigonometric Ratio:
   • If you know/need the opposite side and the hypotenuse, use sine (sin θ = Opposite/Hypotenuse).
   • If you know/need the adjacent side and the hypotenuse, use cosine (cos θ = Adjacent/Hypotenuse).
   • If you know/need the opposite side and the adjacent side, use tangent (tan θ = Opposite/Adjacent).
   • Tip: Remember SOH CAH TOA!
4. Formulate the Equation: Substitute the values from your diagram into the chosen trigonometric ratio.
5. Solve for the Unknown: Use algebraic methods to find the value of the unknown height or distance. Often, you'll need the values of specific angles (like 30°, 45°, 60°).

Important Considerations:

• Observer's Height: Unless specified, assume the observer's height is negligible (i.e., the angle is measured from ground level). If given, you must add or subtract it from the calculated height.
• Parallel Lines: Horizontal lines of sight (for angles of depression) are parallel to the ground. This helps equate alternate interior angles.

3. Worked Example

Problem: A tower stands vertically on the ground. From a point on the ground, which is 15 meters away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower.

Solution:

1. Draw a Diagram:

   • Let AB be the tower (height = h meters).
   • Let C be the point on the ground 15 m away from the foot of the tower (B).
   • The angle of elevation from C to A is 60°.
   • We have a right-angled triangle ABC, with the right angle at B.

2. Identify Knowns and Unknowns:

   • Adjacent side (BC) = 15 m
   • Angle (C) = 60°
   • Opposite side (AB) = h (unknown height)

3. Choose the Correct Trigonometric Ratio:
   We have the opposite side (h) and the adjacent side (15 m). The ratio connecting these two is tangent.
   tan θ = Opposite / Adjacent

4. Formulate the Equation:
   tan 60° = AB / BC
   tan 60° = h / 15

5. Solve for the Unknown:
   We know that tan 60° = √3.
   √3 = h / 15
   h = 15√3 meters

   If you need a numerical value, use √3 ≈ 1.732:
   h ≈ 15 * 1.732
   h ≈ 25.98 meters

   So, the height of the tower is approximately 25.98 meters.

4. Key Takeaways

• Always start by drawing a clear, labelled diagram; it simplifies everything.
• Understand the difference between the angle of elevation (looking up) and the angle of depression (looking down).
• Remember that the angle of elevation from A to B is equal to the angle of depression from B to A.
• Use SOH CAH TOA to correctly choose sine, cosine, or tangent based on the sides you know/need.
• Be familiar with the trigonometric values for common angles (0°, 30°, 45°, 60°, 90°).

Common Mistakes to Avoid:
* Incorrectly drawing the diagram: This is the most common mistake and leads to entirely wrong answers.
* Confusing angle of elevation with depression: Always draw the horizontal line first.
* Using the wrong trigonometric ratio: Double-check if you need sin, cos, or tan for the given sides.
* Not accounting for the observer's height: If given, don't forget to add or subtract it from the final calculated height.

5. Now Try It

A ladder 15m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, find the height of the wall. Draw the diagram, identify the right triangle, choose the correct ratio, and solve for the height. You should get a height of 7.5 meters.