Cliste School
Get started

Measurement, Space & Statistics

Trigonometry: Sine and Cosine Rules

Year 10

  • ✓By the end of this lesson students will be able to apply 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 apply 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 sine formula.
  • ✓By the end of this lesson students will be able to solve practical problems involving the Sine and Cosine Rules.

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 side and its opposite angle, and one other piece of information (either another side or another angle). It is particularly useful when you have Angle-Side-Angle (ASA), Angle-Angle-Side (AAS) to find a side, or Side-Side-Angle (SSA) to find an angle.

For a triangle with sides a, b, c and opposite angles A, B, C respectively:\nTo find a side: a/sin A = b/sin B = c/sin C\nTo find an angle: sin A/a = sin B/b = sin C/c
The Cosine Rule

The Cosine Rule is used to find unknown sides or angles in any triangle when the Sine Rule cannot be applied. It is particularly useful when you have Side-Angle-Side (SAS) to find a side, or Side-Side-Side (SSS) to find an angle.

For a triangle with sides a, b, c and opposite angles A, B, C respectively:\nTo find a side: a² = b² + c² - 2bc cos A\nSimilarly: b² = a² + c² - 2ac cos B\nAnd: c² = a² + b² - 2ab cos C\n\nTo find an angle (rearranged): cos A = (b² + c² - a²) / (2bc)\nSimilarly: cos B = (a² + c² - b²) / (2ac)\nAnd: cos C = (a² + b² - c²) / (2ab)
Area of a Non-Right-Angled Triangle

The area of any triangle can be calculated if you know the lengths of two sides and the measure of the included angle (the angle between those two sides).

Area = ½ab sin C\nSimilarly: Area = ½bc sin A\nAnd: Area = ½ac sin B

Key facts to remember

  • 1The Sine Rule: a/sin A = b/sin B = c/sin C (or its reciprocal) is used when you know an angle and its opposite side, plus one other side or angle.
  • 2The Cosine Rule: a² = b² + c² - 2bc cos A (and its variations) is used when you know two sides and the included angle (SAS) to find a side.
  • 3The Cosine Rule (rearranged): cos A = (b² + c² - a²) / (2bc) (and its variations) is used when you know all three sides (SSS) to find an angle.
  • 4The Area of a triangle: Area = ½ab sin C is used when you know two sides and the included angle.
  • 5Angles are typically denoted by capital letters (A, B, C) and the side opposite an angle by the corresponding lowercase letter (a, b, c).
  • 6Always ensure your calculator is in DEGREE mode when performing trigonometric calculations.
  • 7The sum of angles in any triangle is 180°.

Worked examples

Example 1

In triangle ABC, angle A = 45°, angle B = 60° and side a = 10 cm. Find the length of side b, correct to two decimal places.

I1. Identify the knowns: A = 45°, B = 60°, a = 10 cm.
II2. Identify the unknown: b.
III3. Choose the appropriate rule: We have an angle and its opposite side (A and a), and another angle (B), so the Sine Rule is suitable.
IV4. Write down the Sine Rule for sides: a/sin A = b/sin B.
V5. Substitute the known values: 10/sin 45° = b/sin 60°.
VI6. Rearrange to solve for b: b = (10 × sin 60°) / sin 45°.
VII7. Calculate the value: b = (10 × 0.8660...) / 0.7071... = 8.660... / 0.7071... = 12.2474...
VIII8. Round to two decimal places.

Answer

b ≈ 12.25 cm

Always ensure your calculator is in DEGREE mode for trigonometry problems.

Example 2

In triangle PQR, p = 7 m, q = 9 m and r = 12 m. Find the measure of angle P, correct to one decimal place.

I1. Identify the knowns: p = 7 m, q = 9 m, r = 12 m.
II2. Identify the unknown: angle P.
III3. Choose the appropriate rule: We have all three sides (SSS), so the Cosine Rule for angles is suitable.
IV4. Write down the Cosine Rule for angle P: cos P = (q² + r² - p²) / (2qr).
V5. Substitute the known values: cos P = (9² + 12² - 7²) / (2 × 9 × 12).
VI6. Calculate the numerator: 81 + 144 - 49 = 176.
VII7. Calculate the denominator: 2 × 9 × 12 = 216.
VIII8. So, cos P = 176 / 216 = 0.8148...
99. Find angle P using the inverse cosine function: P = cos⁻¹(0.8148...).
1010. Calculate the value: P ≈ 35.436...
1111. Round to one decimal place.

Answer

P ≈ 35.4°

Remember to use the inverse cosine function (cos⁻¹) to find the angle.

Example 3

A triangular garden has sides of length 8 m and 11 m, with the included angle between them being 70°. Calculate the area of the garden, correct to one decimal place.

I1. Identify the knowns: Two sides (a = 8 m, b = 11 m) and the included angle (C = 70°).
II2. Identify the unknown: Area.
III3. Choose the appropriate formula: Area = ½ab sin C.
IV4. Substitute the known values: Area = ½ × 8 × 11 × sin 70°.
V5. Calculate the product: Area = 44 × sin 70°.
VI6. Calculate sin 70°: sin 70° ≈ 0.93969...
VII7. Multiply: Area = 44 × 0.93969... = 41.346...
VIII8. Round to one decimal place.

Answer

Area ≈ 41.3 m²

The included angle is crucial for this formula. Ensure the angle is between the two given sides.

Common mistakes

  • ✗Using the Sine Rule when the Cosine Rule is required (e.g., for SSS or SAS cases).
  • ✗Using the Cosine Rule when the Sine Rule is simpler and more direct (e.g., for AAS or ASA cases).
  • ✗Not using the 'included angle' for the area formula or the Cosine Rule for finding a side.
  • ✗Incorrectly rearranging the Cosine Rule formula, especially when solving for an angle.
  • ✗Forgetting to use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) when finding an angle.
  • ✗Rounding intermediate steps too early, leading to inaccuracies in the final answer.

Exam tips

  • ★Draw a clear diagram and label all known and unknown sides and angles. This helps in choosing the correct rule.
  • ★Identify what information you have (e.g., ASA, AAS, SAS, SSS) to determine whether to use the Sine Rule, Cosine Rule, or Area formula.
  • ★Write down the formula you are using before substituting values. This helps in showing your working and can earn method marks.
  • ★Check your answer for reasonableness. For example, the longest side should be opposite the largest angle, and vice-versa.

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