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Trigonometry – Heights & Distances
📐 Trigonometry – Heights & Distances
Clear concepts • Exam-ready notes • Easy diagrams
1️⃣ Introduction
Heights and Distances is an important application of trigonometry used to calculate the height of objects or the distance between objects which cannot be measured directly.
👉 This topic is based on right-angled triangles and
trigonometric ratios.
2️⃣ Line of Sight
The line of sight is the straight line drawn from the observer’s eye to the object being viewed.
3️⃣ Angles Used
Angle of Elevation
The angle formed between the horizontal line and the line of sight when the object is above the observer.
Angle of Depression
The angle formed between the horizontal line and the line of sight when the object is below the observer.
⭐ Angle of elevation = Angle of depression
4️⃣ Trigonometric Ratios (SOH–CAH–TOA)
| Ratio | Formula |
|---|---|
| sin θ | Opposite / Hypotenuse |
| cos θ | Adjacent / Hypotenuse |
| tan θ | Opposite / Adjacent |
5️⃣ Standard Trigonometric Values
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
6️⃣ Steps to Solve Problems
- Draw a neat labelled diagram
- Identify the right-angled triangle
- Mark the given angle
- Choose the correct trigonometric ratio
- Substitute values and solve
- Write the answer with proper units
7️⃣ Worked Example
The angle of elevation of the top of a tower is 30°. The distance from the tower is 20 m.
tan 30° = h / 20
1/√3 = h / 20
h = 20/√3 m
1/√3 = h / 20
h = 20/√3 m
⭐ Exam Tips
- Always draw a diagram
- Choose correct trigonometric ratio
- Keep units consistent
- Mention the formula used
❌ Common Mistakes
- Confusing elevation and depression
- Using wrong trigonometric ratio
- Forgetting horizontal distance
- Missing units in the answer