Right-Angled Triangle
6.2 Right-angled triangles – Cheat Sheet
Trigonometric Ratios
For a right-angled triangle with angle θ:
- Sine: sin(θ) = opposite / hypotenuse
- Cosine: cos(θ) = adjacent / hypotenuse
- Tangent: tan(θ) = opposite / adjacent
Formulas Table
| Find | Formula | Notes |
|---|---|---|
| Opposite side | opposite = hypotenuse × sin(θ) | Use when hypotenuse & angle are known |
| Adjacent side | adjacent = hypotenuse × cos(θ) | Use when hypotenuse & angle are known |
| Opposite side | opposite = adjacent × tan(θ) | Use when adjacent & angle are known |
| Adjacent side | adjacent = opposite ÷ tan(θ) | Use when opposite & angle are known |
| Hypotenuse | hypotenuse = opposite ÷ sin(θ) | Use when opposite & angle are known |
| Hypotenuse | hypotenuse = adjacent ÷ cos(θ) | Use when adjacent & angle are known |
| Angle θ (sin) | θ = sin⁻¹(opposite ÷ hypotenuse) | Answer in degrees (°) |
| Angle θ (cos) | θ = cos⁻¹(adjacent ÷ hypotenuse) | Answer in degrees (°) |
| Angle θ (tan) | θ = tan⁻¹(opposite ÷ adjacent) | Answer in degrees (°) |
Pythagoras’ Theorem Review
a² + b² = c²
- a, b = legs (shorter sides)
- c = hypotenuse (longest side)
Angles of Elevation & Depression
- Angle of elevation: measured upwards from horizontal to the object
- Angle of depression: measured downwards from horizontal to the object
- Use trigonometric ratios to calculate missing sides/angles
Worked Examples
| Example | Given | Workings | Answer |
|---|---|---|---|
| 1 | Hypotenuse = 10 m, θ = 30° → Opposite side | opposite = 10 × sin(30°) = 10 × 0.5 | 5.0 m |
| 2 | Adjacent = 12 cm, θ = 40° → Hypotenuse | hypotenuse = 12 ÷ cos(40°) ≈ 12 ÷ 0.7660 | ≈ 15.66 cm |
| 3 | Opposite = 7 m, Adjacent = 9 m → Angle θ | θ = tan⁻¹(7 ÷ 9) = tan⁻¹(0.7777) | ≈ 37.9° |
| 4 (Elevation) | Observer at ground level, building height 20 m, distance 25 m | θ = tan⁻¹(20 ÷ 25) | ≈ 38.7° |