Geometry & measures

Sine and Cosine Rules

Year 10 · Year 11

✓By the end of this lesson students will be able to use the Sine Rule to find unknown sides and angles in non-right-angled triangles.
✓By the end of this lesson students will be able to use the Cosine Rule to find unknown sides and angles in non-right-angled triangles.
✓By the end of this lesson students will be able to calculate the area of a non-right-angled triangle using the formula Area = ½ab sin C.
✓By the end of this lesson students will be able to choose the appropriate rule (Sine Rule, Cosine Rule, or Area Rule) to solve problems involving non-right-angled triangles.

Key concepts

The Sine Rule

The Sine Rule is used to find unknown sides or angles in any triangle (not just right-angled triangles) when you have a matching pair of an angle and its opposite side, plus one other piece of information. This typically means you have Angle-Angle-Side (AAS), Angle-Side-Angle (ASA), or Side-Side-Angle (SSA, though this can sometimes lead to an ambiguous case with two possible triangles).

a/sin A = b/sin B = c/sin C (to find a side) OR sin A/a = sin B/b = sin C/c (to find an angle)

The Cosine Rule

The Cosine Rule is used in non-right-angled triangles when the Sine Rule cannot be applied. It is used in two main scenarios: 1. To find an unknown side when you know two sides and the included angle (Side-Angle-Side, SAS). 2. To find an unknown angle when you know all three sides (Side-Side-Side, SSS).

a² = b² + c² - 2bc cos A (to find a side) OR cos A = (b² + c² - a²) / 2bc (to find an angle)

Area of a Triangle (½ab sin C)

This formula allows you to calculate the area of any triangle when you know the lengths of two sides and the size of the included angle (the angle between those two sides). This is often referred to as the Side-Angle-Side (SAS) case for area.

Area = ½ab sin C

Key facts to remember

1The Sine Rule: a/sin A = b/sin B = c/sin C (or its reciprocal) is used when you have a side and its opposite angle, plus one other side or angle.
2The Cosine Rule for sides: a² = b² + c² - 2bc cos A is used when you know two sides and the included angle (SAS) and want to find the third side.
3The Cosine Rule for angles: cos A = (b² + c² - a²) / 2bc is used when you know all three sides (SSS) and want to find an angle.
4The Area of a Triangle formula: Area = ½ab sin C is used when you know two sides and the included angle (SAS).
5Always label the vertices of a triangle with capital letters (A, B, C) and the sides opposite those vertices with corresponding lowercase letters (a, b, c).
6Ensure your calculator is always in DEGREE mode when working with trigonometry.

Worked examples

Example 1

In triangle ABC, angle A = 42°, angle B = 65°, and side a = 10 cm. Find the length of side b, correct to 3 significant figures.

IIdentify knowns: A = 42°, B = 65°, a = 10 cm. We need to find b.

IISince we have a matching pair (angle A and side a) and another angle (B), we can use the Sine Rule to find side b.

IIIWrite down the Sine Rule for sides: a/sin A = b/sin B

IVSubstitute the known values: 10/sin 42° = b/sin 65°

VRearrange to solve for b: b = (10 × sin 65°) / sin 42°

VICalculate the value: b = (10 × 0.9063...) / 0.6691...

VIIb ≈ 13.545 cm

Answer

b = 13.5 cm (3 s.f.)

Ensure your calculator is in DEGREE mode.

Example 2

In triangle PQR, side p = 8 cm, side q = 11 cm, and side r = 15 cm. Find the size of angle P, correct to 1 decimal place.

IIdentify knowns: p = 8 cm, q = 11 cm, r = 15 cm. We need to find angle P.

IISince we know all three sides (SSS), we use the Cosine Rule to find an angle.

IIIWrite down the Cosine Rule for angle P: cos P = (q² + r² - p²) / 2qr

IVSubstitute the known values: cos P = (11² + 15² - 8²) / (2 × 11 × 15)

VCalculate the numerator and denominator: cos P = (121 + 225 - 64) / 330

VIcos P = 282 / 330

VIIcos P = 0.8545...

VIIIFind angle P by taking the inverse cosine: P = cos⁻¹(0.8545...)

9P ≈ 31.29°

Answer

Angle P = 31.3° (1 d.p.)

Remember to use cos⁻¹ (or arccos) to find the angle after calculating its cosine value.

Example 3

Triangle XYZ has sides x = 14 cm, y = 18 cm and the included angle Z = 80°. a) Calculate the area of triangle XYZ, correct to 3 significant figures. b) Calculate the length of side z, correct to 3 significant figures.

Ia) Calculate the area:

IIIdentify knowns: x = 14 cm, y = 18 cm, included angle Z = 80°. We need the area.

IIISince we have two sides and the included angle (SAS), we use the Area Rule: Area = ½xy sin Z

IVSubstitute the values: Area = ½ × 14 × 18 × sin 80°

VCalculate: Area = 126 × sin 80°

VIArea = 126 × 0.9848...

VIIArea ≈ 124.08 cm²

VIIIb) Calculate the length of side z:

9Identify knowns: x = 14 cm, y = 18 cm, included angle Z = 80°. We need side z.

10Since we have two sides and the included angle (SAS) and need the opposite side, we use the Cosine Rule for sides: z² = x² + y² - 2xy cos Z

11Substitute the values: z² = 14² + 18² - (2 × 14 × 18 × cos 80°)

12Calculate: z² = 196 + 324 - (504 × cos 80°)

13z² = 520 - (504 × 0.1736...)

14z² = 520 - 87.49...

15z² = 432.50...

16Find z by taking the square root: z = √432.50...

17z ≈ 20.79 cm

Answer

a) Area = 124 cm² (3 s.f.) b) z = 20.8 cm (3 s.f.)

This example combines two rules, which is common in higher-tier exam questions.

Common mistakes

✗Using the wrong rule: Students often try to use the Sine Rule when the Cosine Rule is required (e.g., for SSS or SAS for a side) or vice-versa.
✗Incorrectly pairing sides and angles: Forgetting that 'a' must be opposite 'A', 'b' opposite 'B', etc., leading to incorrect substitutions.
✗Calculator mode errors: Not checking that the calculator is in DEGREE mode, resulting in incorrect trigonometric values.
✗Algebraic errors: Incorrectly rearranging the formulas, especially the Cosine Rule for angles, or forgetting to take the square root when finding a side.
✗Ambiguous case of the Sine Rule: When finding an angle using the Sine Rule, if sin X = k, there might be two possible angles (X and 180°-X). At GCSE, questions usually avoid this or provide diagrams where the acute/obtuse nature is clear.

Exam tips

★Draw a clear diagram: Even if one is provided, redraw it and label all known and unknown sides/angles. This helps in visualising the problem and choosing the correct rule.
★Identify the given information: Determine if you have SAS, SSS, AAS, or ASA. This will guide you to the correct formula (Sine Rule, Cosine Rule, or Area Rule).
★Write down the formula: Always state the formula you are using before substituting values. This shows your method and can earn marks even if there's a calculation error.
★Show full working: Present your steps clearly and logically. This makes it easier for the examiner to follow your reasoning and award method marks.
★Check your answer: Does your answer make sense in the context of the triangle? For example, the longest side should be opposite the largest angle. Give answers to the specified degree of accuracy and include units.

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum