Understanding the trigonometric rules of the Sine and Cosine is essential for mastering non-right-angled triangle problems in GCSE Maths. These rules are especially useful in higher-tier questions that require calculating unknown sides or angles when right-angle trigonometry doesn’t apply. This guide will help you know when and how to use each rule, with formulas, diagrams, worked examples, and exam-focused strategies.
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🔹Trigonometric Sine Rule – When and How to Use It
The Sine Rule is used in non-right-angled triangles when you have:
- A pair of an angle and its opposite side
- Another known angle or side
✅ Sine Rule Formula:
a / sin(A) = b / sin(B) = c / sin(C)Where:
- a, b, c are sides
- A, B, and C are the opposite angles
📌 Use Sine Rule When:
- You know two angles and one side (AAS or ASA)
- You know two sides and a non-included angle (SSA)
🧮 Example:
In triangle ABC:
- Angle A = 40°, Angle B = 60°, side a = 7 cm
- Use the sine rule to find side b
b / sin(60°) = 7 / sin(40°)
b = [7 × sin(60°)] / sin(40°)
b ≈ 10.17 cm⚠️ Ambiguous Case:
Be cautious with SSA cases; they may yield two solutions (or none).
🔸Trigonometric Cosine Rule – When and How to Use It
The Cosine Rule is used when:
- You know two sides and the included angle (SAS)
- You know all three sides (SSS) and want to find an angle
✅ Cosine Rule Formulas:
To find a side:
a² = b² + c² − 2bc × cos(A)To find an angle:
cos(A) = (b² + c² − a²) / (2bc)📌 Use the Cosine Rule When:
- You’re given SAS or SSS information
- You’re solving for a side or angle not opposite to a known angle
🧮 Example:
Given:
- a = ?, b = 7 cm, c = 9 cm, angle A = 60°
a² = 7² + 9² − 2×7×9×cos(60°)
a² = 49 + 81 − 126×0.5 = 130 − 63 = 67
a = √67 ≈ 8.19 cm🔸 Choosing Between Sine and Cosine Rule
| What You Know | Use This Rule |
|---|---|
| Two angles + one side | Sine Rule |
| Two sides + included angle | Cosine Rule |
| Two sides + non-included angle | Sine Rule (ambiguous case) |
| All three sides | Cosine Rule |
Always start by sketching the triangle and labelling all known values clearly.
🧠 Exam Tips for Trig Rules
- Always label your triangle first
- Round answers to 1 decimal place unless stated
- Use radians or degrees consistently (use DEG mode in calculator for GCSE)
- Watch for ambiguous SSA cases
- Check if the triangle is right-angled before using these rules
📝 Practice Questions
- In triangle ABC, angle A = 50°, angle B = 65°, and side a = 8 cm. Find side b.
- Given a triangle with sides 6 cm, 10 cm, the angle between them is 45°. Find the third side.
- In triangle PQR, sides p = 10 cm, q = 8 cm, r = 7 cm. Find angle P.
- Solve triangle XYZ given side x = 9 cm, angle X = 35°, angle Y = 70°.
✅ Final Thoughts
The Sine and Cosine Rules are vital tools for handling more advanced non-right-angled triangle problems. Whether you’re given ambiguous side-angle combinations or complete sets of sides, knowing when and how to apply the correct formula will give you a significant edge in exams.
For expert-guided practice, revision worksheets, and one-on-one help, visit GCSE Maths Tutor and strengthen your trigonometry confidence.
