Why Heights and Distances Matter in Real Life
Ever wondered how engineers figure out the height of a tower without climbing it? That's exactly what trigonometry helps us do.
💡 In Simple Words: When you see a slanted line (like a ladder) and you know either its angle or length, you can use trigonometric ratios to find the missing height or distance. It’s like measuring a tree’s height by looking at the shadow.
Key Ideas You Need
Trigonometric ratios
Sine (sin) = opposite side ÷ hypotenuse. Think of it as the height divided by the slanted side.
Cosine (cos) = adjacent side ÷ hypotenuse. Like the ground distance divided by the slanted side.
Tangent (tan) = opposite side ÷ adjacent side. It compares height directly to ground distance.
When to use which ratio
- If you know the angle and the hypotenuse, use sin or cos.
- If you know the angle and one side that isn’t the hypotenuse, use tan.
- When the problem involves a right‑angled triangle, the ratios work perfectly.
Step‑by‑Step Process (Flowchart)
Worked Example 1
Problem: A kite is flying at a height of 30 m. The string makes an angle of 45° with the ground. Find the length of the string.
Solution:
- Draw the right‑angled triangle: height = 30 m (opposite), angle = 45°, string = hypotenuse.
- Use the sine ratio: sin 45° = opposite ÷ hypotenuse.
- sin 45° = √2/2 ≈ 0.707. So 0.707 = 30 ÷ hypotenuse.
- Hypotenuse = 30 ÷ 0.707 ≈ 42.4 m.
The string is about 42 m long.
Worked Example 2
Problem: From a point on the ground, the angle of elevation to the top of a building is 30°. If you walk 20 m towards the building and the angle becomes 45°, how tall is the building?
Solution:
- Let the original distance from the building be x metres.
- First triangle: tan 30° = height ÷ x → height = x·tan 30° = x·(1/√3).
- After walking 20 m, new distance = x − 20. Second triangle: tan 45° = height ÷ (x − 20) → height = x − 20.
- Set the two expressions for height equal: x·(1/√3) = x − 20.
- Solve: x/√3 = x − 20 → multiply by √3: x = √3·x − 20√3 → bring terms: x − √3·x = ‑20√3 → x(1‑√3) = ‑20√3 → x = (20√3)/(√3 ‑ 1) ≈ 44.1 m.
- Height = x − 20 ≈ 24.1 m.
So the building is about 24 m tall.
Quick Reference Table
| Known | Need | Use Ratio |
|---|---|---|
| Angle + hypotenuse | Opposite (height) or adjacent (distance) | sin or cos |
| Angle + opposite | Adjacent or hypotenuse | tan (or rearranged sin/cos) |
| Angle + adjacent | Opposite or hypotenuse | tan (or rearranged sin/cos) |
📝 Likely Exam Questions
- From a point 50 m away from a hill, the angle of elevation to its peak is 30°. Find the height of the hill.
- A tree casts a shadow 12 m long when the sun’s elevation angle is 40°. Calculate the tree’s height.
- The angle of depression from the top of a 25 m high tower to a boat is 20°. How far is the boat from the base of the tower?
- A building is observed from two points on the same straight line. From the farther point, the angle of elevation is 25°. After moving 15 m towards the building, the angle becomes 35°. Find the building’s height.
Model answers: use tan θ = opposite/adjacent, solve simple equations, round to nearest metre unless otherwise asked.