Angles of Elevation and Depression | Trigonometry

🎯 Learning Objectives

distinguish elevation and depression;
draw horizontal reference lines;
translate word problems into right triangles;
calculate heights and distances.

Introduction

This lesson develops Angles of Elevation and Depression from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.

Key Notes

Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.

Visualising the Situation

The angle of elevation is measured upward from the observer’s horizontal line.

Worked Example

An observer stands 25 m from a tower. The angle of elevation is 38°.

tan38°=h/25

h=25tan38°

h≈19.5 m

Practice Questions

A tree is viewed from 18 m away at 42°. Find its height.
From a 30 m cliff, the angle of depression to a boat is 25°. Find the horizontal distance.
Explain why the angle of depression equals the angle of elevation.

✅ Check Your Work

Show answers and working

h=18tan42°≈16.2 m.
tan25°=30/d, so d=30/tan25°≈64.3 m.
The two horizontal lines are parallel, so the angles are alternate angles.

🧭 Interactive Lab: Elevation and Depression Explorer

Change the values or make a choice and watch the mathematics respond.

Horizontal distance: 30 m

Vertical height: 20 m

Summary

📘 Remember

Label the triangle before choosing a formula.
Use degree mode unless the question says otherwise.
Keep full calculator values until the final rounding step.

⚠️ Common Mistakes

Using the wrong reference angle.
Rounding too early.
Forgetting units or the degree symbol.

⭐ Exam Tips

Write the formula before substituting.
Sketch and label any diagram that is not already provided.
Check whether the answer is sensible for the shape.

🌍 Did You Know?

Surveyors, engineers, pilots and architects use trigonometry to calculate distances that are difficult or unsafe to measure directly.

Common Mistakes

Choosing a rule before identifying what the question describes.
Skipping working or changing notation part-way through a solution.
Accepting an answer without checking its size, sign or units.

Exam Focus

Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.