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Trigonometry: cosine rule — KCSE Mathematics

KCSE Mathematics · 117 practice questions · 3 syllabus objectives · 3 revision lessons

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Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

State the cosine rule: c² = a² + b² – 2ab cos C (and its rearrangement) and identify when to apply it (given SAS or SSS)

Apply the cosine rule to calculate unknown sides and angles in triangles; use the formula A = ½ab sin C for area

Trigonometry: cosine rule

Revision Notes

Concise lesson notes for Trigonometry: cosine rule, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Understanding the Cosine Rule

The cosine rule is a fundamental relationship in trigonometry, useful for solving triangles. It states that for any triangle with sides a, b, and c opposite to angles A, B, and C respectively:

c² = a² + b² - 2ab cos C

This formula helps to find the length of a side when two sides and the included angle (SAS) are known or to find an angle when all three sides (SSS) are known.

Rearrangement

The cosine rule can also be rearranged to find the cosine of an angle:

  • cos C = (a² + b² - c²) / (2ab)

When to Apply the Cosine Rule:

  • Use when you have two sides and the included angle (SAS).
  • Use when you have all three sides (SSS).

Understanding when to apply this rule is crucial for solving problems involving triangles effectively.

Key points to remember

  • Cosine rule: c² = a² + b² - 2ab cos C.
  • Rearrangement: cos C = (a² + b² - c²) / (2ab).
  • Apply for SAS: two sides and included angle.
  • Apply for SSS: all three sides known.
  • Essential for solving non-right triangles.

Worked example

Question: In triangle ABC, if a = 5 cm, b = 7 cm, and angle C = 60°, find side c.

  • Use cosine rule: c² = a² + b² - 2ab cos C.
  • c² = 5² + 7² - 2(5)(7)(0.5).
  • c² = 25 + 49 - 35 = 39.
  • c = √39 ≈ 6.24 cm.

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More lessons in this topic

Lesson 2: Applying the Cosine Rule in Triangles

Objective: Apply the cosine rule to calculate unknown sides and angles in triangles; use the formula A = ½ab sin C for area

The cosine rule is a powerful tool in trigonometry, used to find unknown sides and angles in any triangle. The formula is given by:
c² = a² + b² - 2ab cos(C),
where:

  • c is the side opposite angle C,
  • a and b are the other two sides,
  • C is the angle between sides a and b.
    To find an unknown angle, rearrange the formula:
    C = cos⁻¹((a² + b² - c²) / (2ab)).

To calculate the area of a triangle, use the formula:
Area = ½ab sin(C),
where a and b are two sides, and C is the included angle.

Example:
Given a triangle with sides a = 5 cm, b = 7 cm, and angle C = 60°, find side c and the area.

  1. Calculate c:
    • c² = 5² + 7² - 2(5)(7)cos(60°)
    • c² = 25 + 49 - 35 = 39
    • c = √39 ≈ 6.24 cm
  2. Calculate the area:
    • Area = ½(5)(7)sin(60°) ≈ 12.1 cm².
  • Use c² = a² + b² - 2ab cos(C) for sides.
  • Rearrange to find angles using C = cos⁻¹(...).
  • Calculate area with Area = ½ab sin(C).
  • Ensure angle C is in degrees for calculations.
  • Check units and round off your answers appropriately.

Find side c in a triangle with a = 8, b = 6, C = 45°.

  • c² = 8² + 6² - 2(8)(6)cos(45°)
  • c² = 64 + 36 - 96(0.707) ≈ 34.67
  • c = √34.67 ≈ 5.88.
Lesson 3: Understanding the Cosine Rule

Objective: Trigonometry: cosine rule

The Cosine Rule is essential in solving triangles, particularly when you know two sides and the included angle or all three sides. It states that for any triangle with sides a, b, and c opposite to angles A, B, and C respectively, the formula is:

c² = a² + b² - 2ab * cos(C)

This formula allows you to find the length of a side or the measure of an angle. To apply the Cosine Rule:

  1. Identify the sides and the angle involved.
  2. Substitute the known values into the formula.
  3. Rearrange as necessary to solve for the unknown.

Example: Given a triangle with sides a = 5 cm, b = 7 cm, and angle C = 60°:

  • Calculate side c.

Using the formula: c² = 5² + 7² - 2(5)(7) * cos(60°) c² = 25 + 49 - 70 * 0.5 c² = 25 + 49 - 35 c² = 39 c = √39 ≈ 6.24 cm.

Thus, the length of side c is approximately 6.24 cm.

  • Cosine Rule relates sides and angles in triangles.
  • Formula: c² = a² + b² - 2ab * cos(C).
  • Use when given two sides and included angle.
  • Rearrange the formula to find unknowns.
  • Applies to all triangle types, not just right-angled.

Given triangle ABC with sides a = 8 cm, b = 6 cm, and angle C = 45°, find side c.

c² = 8² + 6² - 2(8)(6) * cos(45°) c² = 64 + 36 - 96 * 0.707 c² ≈ 64 + 36 - 67.3 c² ≈ 32.7 c ≈ √32.7 ≈ 5.72 cm.

Sample Questions

Read 3 questions and answers free. Sign up to access all 117 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER3 marks

State the cosine rule and explain when it is applicable in triangle measurements. (3 marks)

Answer & marking scheme

Part (a) — 1 mark
c² = a² + b² − 2ab cos C (1 mk)
Part (b) — 2 marks
Applicable when two sides and the included angle (SAS) are known (1 mk)
Also applicable when all three sides (SSS) are known (1 mk)
2
easySHORT ANSWER4 marks

In triangle XYZ, the lengths of sides XY and XZ are 8 cm and 6 cm respectively, with an included angle ∠Y measuring 45°. Determine the area of triangle XYZ using the formula A = ½ab sin C. (4 marks)

Answer & marking scheme

Part (a) — 4 marks
Identify a = 8 cm, b = 6 cm, and C = 45° (1 mk)
Use the area formula: A = ½(8)(6)sin(45°) (1 mk)
Calculate the sine of 45°, which is √2/2 (1 mk)
Final area calculation: A = 24√2/2 = 12√2 cm² (approximately 16.97 cm²) (1 mk)
3
easySHORT ANSWER3 marks

In triangle PQR, side PQ measures 7 cm, side PR measures 5 cm, and the included angle ∠Q is 60°. Calculate the length of side QR using the cosine rule. (3 marks)

Answer & marking scheme

Part (a) — 3 marks
Apply the cosine rule: QR² = PQ² + PR² - 2(PQ)(PR)cos(Q) (1 mk)
Substitute values: QR² = 7² + 5² - 2(7)(5)cos(60°) (1 mk)
Calculate QR and provide the final answer: QR = 6.32 cm (approximately) (1 mk)
4

State the rearranged form of the cosine rule and describe its use in calculating an angle in triangle XYZ. (4 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Trigonometry: cosine rule" cover?

Trigonometry: cosine rule covers State the cosine rule: c² = a² + b² – 2ab cos C (and its rearrangement) and identify when to apply it (given SAS or SSS); Apply the cosine rule to calculate unknown sides and angles in triangles; use the formula A = ½ab sin C for area; Trigonometry: cosine rule, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Trigonometry: cosine rule?

HighMarks has 117 Trigonometry: cosine rule practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Trigonometry: cosine rule for the KCSE exam?

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