Why Trigonometry Matters
Trigonometry is one of the most heavily tested topics in both E-Maths and A-Maths at the O-Level. It connects algebra, geometry, and real-world problem-solving, appearing in questions about heights and distances, bearings, and wave motion. A solid foundation here is essential.
Part 1: Right-Angled Triangles — SOH CAH TOA
For any right-angled triangle, the three basic trigonometric ratios are:
- sin θ = Opposite / Hypotenuse
- cos θ = Adjacent / Hypotenuse
- tan θ = Opposite / Adjacent
The mnemonic SOH CAH TOA helps you recall these instantly. Always identify the hypotenuse first (the side opposite the right angle), then label the other two sides relative to your chosen angle.
Worked Example
In a right-angled triangle, one angle is 35° and the hypotenuse is 10 cm. Find the side opposite the 35° angle.
Using sin: sin 35° = Opposite / 10 → Opposite = 10 × sin 35° ≈ 5.74 cm
Part 2: Angles of Elevation and Depression
These are practical applications of right-angled trigonometry:
- Angle of elevation: The angle measured upward from the horizontal to an object above.
- Angle of depression: The angle measured downward from the horizontal to an object below.
Both are always measured from the horizontal, not from the vertical. This is a common source of errors.
Part 3: Bearings
Bearings measure direction as a three-digit angle clockwise from North.
- North = 000°, East = 090°, South = 180°, West = 270°
- Always draw a clear diagram with North arrows before solving.
- Use alternate angles and co-interior angles (from parallel North lines) to find interior triangle angles.
Part 4: Non-Right-Angled Triangles
For triangles without a right angle, use the Sine Rule or Cosine Rule.
The Sine Rule
a / sin A = b / sin B = c / sin C
Use it when you know: an angle and its opposite side, plus one more angle or side (AAS or ASA situations).
The Cosine Rule
a² = b² + c² − 2bc cos A
Use it when you know: all three sides (SSS), or two sides and the included angle (SAS).
Choosing the Right Formula
| Given Information | Use |
|---|---|
| Right-angled triangle | SOH CAH TOA |
| Two angles + one side (AAS/ASA) | Sine Rule |
| Two sides + non-included angle (SSA) | Sine Rule (watch for ambiguous case) |
| Three sides (SSS) | Cosine Rule |
| Two sides + included angle (SAS) | Cosine Rule |
Area of a Triangle
When you know two sides and the included angle:
Area = ½ ab sin C
This formula is used frequently in exam questions alongside the Sine and Cosine Rules.
Common Mistakes to Avoid
- Using SOH CAH TOA on non-right-angled triangles.
- Confusing which side is opposite a given angle when labelling triangles.
- Forgetting the ambiguous case in the Sine Rule (two possible triangles).
- Rounding intermediate values — always keep full calculator precision until the final answer.
Conclusion
Trigonometry rewards students who take time to draw clear, well-labelled diagrams and systematically identify what information they have. Master each section progressively, from right-angled triangles all the way to the Sine and Cosine Rules, and you'll be well-prepared for any trigonometry question in your O-Level exam.