Sine & Cosine Rules

Solve non-right-angled triangles to find missing sides, angles, and area using a/sinA = b/sinB and a² = b² + c² - 2bc cosA.

Learning Objectives

Label triangles correctly with opposite uppercase angles and lowercase sides
Apply the Sine Rule to find sides or angles given matching pairs
Apply the Cosine Rule when given SAS (two sides and enclosed angle) or SSS (three sides)
Calculate triangle area using ½ ab sin(C)

Concept Explanation

While SOHCAHTOA only works on right-angled triangles, the Sine Rule and Cosine Rule allow you to calculate any missing side or angle in ANY general triangle.

Triangle Labelling: Capital letters (A, B, C) represent vertices/angles. Lowercase letters (a, b, c) represent sides opposite to those angles.

The Sine Rule: Used when you know an opposite side-angle pair. Formula: a / sin A = b / sin B = c / sin C.

The Cosine Rule for Sides: Used when you know two sides and the included angle (SAS): a² = b² + c² - 2bc cos A.

The Cosine Rule for Angles: Rearranged when you know all three sides (SSS): cos A = (b² + c² - a²) / (2bc).

Key Mathematical Formulas

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

a^2 = b^2 + c^2 - 2bc \cos A

\cos A = \frac{b^2 + c^2 - a^2}{2bc}

\text{Area} = \frac{1}{2} ab \sin C

Worked Example

Step-by-Step Solution

Interactive Walkthrough

Problem Statement:

In triangle ABC, side b = 7 cm, side c = 10 cm, and angle A = 60°. Calculate side length a.

b = 7\text{ cm}, \quad c = 10\text{ cm}, \quad A = 60^\circ, \quad a = ?

Step 1: Select the Cosine Rule

We have two sides and the enclosed angle (SAS), so we use a² = b² + c² - 2bc cos A.

a^2 = b^2 + c^2 - 2bc \cos A

Final Answer:

a = 8.89\text{ cm}

Try It Yourself

Calculate the area of a triangle with sides a = 8 cm, b = 11 cm, and enclosed angle C = 30°.

\text{Area} = \frac{1}{2} ab \sin C

Common Student Mistakes

❌ Common Pitfall: Incorrect order of operations in the Cosine Rule

✅ Correction: In a² = b² + c² - 2bc cos A, calculate (2bc cos A) FIRST before subtracting from (b² + c²).

UK Exam Advice & Tips

Memories exact trig values for GCSE Higher: sin(30°)=0.5, cos(60°)=0.5, sin(45°)=√2/2.

Lesson FAQ

What is the ambiguous case of the Sine Rule?

When calculating an angle using the Sine Rule (SSA), sin θ = sin(180° - θ). There can be two valid angles: an acute angle θ or an obtuse angle (180° - θ).

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