Trigonometry appears in every CIE 0580 Paper 4 and Edexcel 4MA1 Paper 2H. It typically carries $10$–$14$ marks. Here is every technique you need, in order of difficulty.
Level 1: Right-Angled Triangles (SOHCAHTOA)
The foundation. Applicable when you have a right angle ($90°$).
| Ratio | Formula | When to Use |
|---|---|---|
| $\sin \theta$ | $\frac{\text{opposite}}{\text{hypotenuse}}$ | Know opposite + hypotenuse |
| $\cos \theta$ | $\frac{\text{adjacent}}{\text{hypotenuse}}$ | Know adjacent + hypotenuse |
| $\tan \theta$ | $\frac{\text{opposite}}{\text{adjacent}}$ | Know opposite + adjacent |
Step-by-step method:
- Label the triangle: hypotenuse (longest side, opposite the right angle), opposite (across from the angle you’re working with), adjacent (next to the angle).
- Choose the ratio that involves the two sides you know (or the side you need).
- Set up the equation and solve.
Example: Find angle $x$ in a right-angled triangle where opposite $= 5$ cm and hypotenuse $= 13$ cm.
$\sin x = \frac{5}{13}$ $x = \sin^{-1}\left(\frac{5}{13}\right)$ $x = 22.6°$ (1 d.p.)
Common mistake: Using the wrong ratio. Always label sides first, then pick the formula.
Level 2: Bearings and Angles of Elevation
Bearings and elevation problems are right-angled triangle questions in disguise.
Bearings:
- Measured clockwise from North
- Always give $3$ figures (e.g., $045°$, not $45°$)
- Draw a North line at every point
Angles of elevation/depression:
- The angle is always measured from the horizontal
- Draw a horizontal line, then the line of sight
Level 3: Sine Rule (Non-Right-Angled Triangles)
Use when you have:
- Two angles and one side, OR
- Two sides and a non-included angle
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Watch out: The ambiguous case (two possible triangles) appears in Extended/Higher only. If $\sin$ gives an obtuse angle possibility, check both solutions.
Level 4: Cosine Rule (Non-Right-Angled Triangles)
Use when you have:
- Two sides and the included angle (SAS) — to find the third side
- Three sides (SSS) — to find an angle
Finding a side: $a^2 = b^2 + c^2 - 2bc \cos A$
Finding an angle: $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$
Tip: If the angle is obtuse, $\cos$ will be negative. Don’t panic — this is correct.
Level 5: Area of a Triangle Using Trigonometry
$\text{Area} = \frac{1}{2}\, ab \sin C$
where $a$ and $b$ are two sides and $C$ is the included angle.
This is faster and more accurate than $\frac{1}{2} \times \text{base} \times \text{height}$ when you don’t have the perpendicular height.
Level 6: 3D Trigonometry (Extended/Higher Only)
3D problems combine right-angled triangles in different planes.
Method:
- Identify the right-angled triangle within the 3D shape.
- Extract it — draw it separately in 2D.
- Solve using SOHCAHTOA or Pythagoras.
- Repeat if you need to chain through multiple triangles.
Common 3D shapes:
- Cuboid: diagonal across a face, space diagonal
- Pyramid: slant height, angle with base
- Triangular prism: angle between edge and base
Exam tip: Always state which triangle you are working in. Write “In triangle ABC…” before showing working.
Quick Reference Card
| Situation | Use |
|---|---|
| Right angle present | SOHCAHTOA |
| Know $2$ angles $+ 1$ side | Sine Rule |
| Know $2$ sides $+$ included angle | Cosine Rule (find side) |
| Know $3$ sides | Cosine Rule (find angle) |
| Need area, know $2$ sides $+$ included angle | $\frac{1}{2}\, ab \sin C$ |
| 3D problem | Extract right-angled triangle, then SOHCAHTOA |
Exam Checklist
- Calculator in degree mode (not radians)
- Angles rounded to $1$ decimal place (unless stated otherwise)
- Lengths rounded as specified
- “Exact answer” → leave as surd or fraction
- Bearing answers always $3$ figures (e.g., $072°$)
Practice Trigonometry Now
- CIE 0580 Trigonometry Practice — interactive questions with instant feedback
- Free Topic Resources
- Weekly Exam Practice Packs
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