If you're studying for your GCSE Maths exam (AQA, Edexcel, or OCR), you'll need to tackle problems involving non-right-angled triangles. That's where the Sine Rule and Cosine Rule come in. These powerful formulas move you beyond basic right-angled trigonometry (SOH CAH TOA) and let you solve any triangle. This guide will walk you through when and how to use each rule, with clear steps and examples.
What Are They For?
First, a quick overview. You use these rules on triangles that are not right-angled. They help you find:
Missing side lengths
- Missing angles
But how do you know which rule to pick? This is the most common stumbling block. Let's clear that up first.
The Golden Rule: Which One to Use?
Follow this simple decision process:
flowchart TD
A["Start: Non-right-angled Triangle Problem"] --> B["What do you know?"]
B --> C["Two angles and one side, AAS or ASA"]
B --> D["Two sides and the included angle SAS"]
B --> E["Three sides SSS"]
C --> F["Use the Sine Rule"]
D --> G["Use the Cosine Rule, to find the missing side"]
E --> H["Use the Cosine Rule, to find an angle"]
F --> I["Can also use Sine Rule for, two sides and a non-included angle SSA, but beware the ambiguous case"]
Key:
AAS/ASA: Two Angles and a Side.
SAS: Two Sides and the Angle between them.
SSS: Three Sides.
- SSA: Two Sides and a non-included angle. Use the Sine Rule, but carefully (more on this later).
The Sine Rule: Step-by-Step
The Sine Rule states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant.
Formula:
asin A = bsin B = csin C
Where (a, b, c) are side lengths, and (A, B, C) are the angles opposite those sides.
When to Use It
Use the Sine Rule when you know:
Two angles and one side (AASASA) – to find a missing side.
- Two sides and a non-included angle (SSA) – to find a missing angle. (Warning: Ambiguous Case!)
Worked Example 1: Finding a Side (AAS)
> In triangle ABC, angle A = 40°, angle B = 75°, and side a = 10 cm. Find the length of side b.
Step 1: Label the triangle.
Side (a = 10) is opposite angle A (40°).
Side (b) is opposite angle B (75°).
Step 2: Write the relevant part of the formula.
We pair (a) with sin A and (b) with sin B.
asin A = bsin B
Step 3: Substitute the known values.
10sin 40° = bsin 75°
Step 4: Solve for (b).
b = 10 × sin 75°/sin 40°
Step 5: Calculate.
b ≈ 10 × 0.9659⁄0.6428 ≈ 9.659⁄0.6428 ≈ 15.0 cm
The Ambiguous Case (SSA)
When you have two sides and a non-included angle (SSA), using the Sine Rule to find an angle can sometimes give two possible answers—an acute angle and its obtuse supplement. This happens because ( sin(θ) = sin(180° - θ) ).
Always check if the triangle is possible: If the side opposite the given angle is the shorter of the two given sides, you may have two possible triangles. Sketch it! Your calculator will give the acute angle first; you must consider if the obtuse angle is also valid given the other angles in the triangle.
The Cosine Rule: Step-by-Step
The Cosine Rule is like a super-powered Pythagoras' theorem for any triangle.
Formulas:
To find a side: ( a² = b² + c² - 2bc cos A )
- To find an angle: ( cos A = b² + c² - a22bc )
When to Use It
Use the Cosine Rule when you know:
Two sides and the included angle (SAS) – to find the third side.
- Three sides (SSS) – to find any angle.
Worked Example 2: Finding a Side (SAS)
> In triangle PQR, side p = 7 m, side q = 9 m, and the included angle R = 60°. Find the length of side r.
Step 1: Identify the knowns.
We have sides (p) and (q) and the angle between them, R. We want side (r), which is opposite angle R.
Step 2: Write the formula for a side.
r² = p² + q² - 2pq cos R
Step 3: Substitute.
r² = 7² + 9² - (2 × 7 × 9 × cos 60°)
r² = 49 + 81 - (126 × 0.5)
r² = 130 - 63
r² = 67
Step 4: Solve.
r = √67 ≈ 8.19 m
Worked Example 3: Finding an Angle (SSS)
> A triangle has sides of length 5 cm, 6 cm, and 7 cm. Find the size of the largest angle.
The largest angle is opposite the longest side (7 cm).
Step 1: Label. Let (a = 7), (b = 5), (c = 6). We want angle A.
Step 2: Write the formula for an angle.
cos A = b² + c² - a22bc
Step 3: Substitute.
cos A = 5² + 6² - 722 × 5 × 6
cos A = 25 + 36 - 4960
cos A = 1260 = 0.2
Step 4: Solve.
A = cos⁻¹(0.2) ≈ 78.5°
Common GCSE Exam Mistakes & How to Avoid Them
Using the wrong rule: Always check the decision flowchart first. Is the angle you know included between the two sides? If yes, use Cosine Rule.
Forgetting to use the inverse function: When finding an angle using the Cosine Rule, you must use the inverse cosine ((cos⁻¹)) on your calculator.
Incorrect calculator mode: Ensure your calculator is in degrees (DEG) mode, not radians (RAD).
Ambiguous case oversight: When using the Sine Rule with SSA, always consider if a second, obtuse angle solution is possible. Does it make the sum of angles in the triangle exceed 180°?
- Rounding too early: Carry through full calculator values until the final answer, then round to 1 decimal place or 3 significant figures as the question requests.
Practice Makes Perfect
The best way to master these rules is through practice. Try these typical GCSE-style questions:
Sine Rule: In triangle XYZ, angle X = 55°, side x = 12 cm, and side y = 15 cm. Find the possible sizes of angle Y.
- Cosine Rule: A triangular field has fences of length 50 m and 80 m meeting at an angle of 110°. How long is the third fence?
Answers (check your working):
Angle Y ≈ 81.3° or its supplement, 98.7°. Both are valid as neither makes the sum of angles exceed 180°.
- Using ( a² = 50² + 80² - 2×50×80×cos110° ), the third side ≈ 111.0 m.
- By following this structured approach—identifying the knowns, choosing the correct rule, and applying the formula carefully—you'll confidently solve any non-right-angled triangle problem that comes up in your GCSE exam. Good luck with your revision!