Note

These were also used for Grade 9 questions in GCSE.

The Sine Rule and Cosine Rule are used to find unknown sides or angles in non-right-angled triangles.

Sine Rule

The Sine Rule states that in any triangle ABC with sides a, b, and c opposite angles A, B, and C respectively:

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

This is useful when you only know one pair (side + angle) and another side or angle. Note: this is not limited to order, i.e., $\frac{a}{\sin A} = \frac{c}{\sin C}$.

Sine Rule Diagram

Example

Cosine Rule

The Cosine Rule states that in any triangle ABC with sides a, b, and c opposite angles A, B, and C respectively:

$c^2 = a^2 + b^2 - 2ab \cdot \cos C$

or rearranged to find the angle:

$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$

This is useful when you know two sides and the angle in between

Requirements

• Two sides
• One included angle (“in the middle”)

Example

Question: Show that $17x^2- 35x - 48 = 0$ (4.a.i.).

Additional Working-Out

Area Without Perpendicular Height

The area of a triangle can also be calculated using the formula:

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

where a and b are two sides of the triangle and C is the included angle between those sides.

Requirements

Same as Cosine Rule.

• Two sides
• One included angle (“in the middle”)