Practice Maths

The Cosine Rule

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Key Terms

Cosine rule (finding a side)
c² = a² + b² − 2ab cos C — a generalisation of Pythagoras’ theorem.
Cosine rule (finding an angle)
cos C = (a² + b² − c²) / (2ab).
When to use
Use when two sides + the included angle (SAS) are given, or all three sides are known (SSS).
Included angle
The angle between the two given sides; must be C in the formula above when using SAS.
Relationship to Pythagoras
If C = 90°, then cos 90° = 0 and the cosine rule reduces to c² = a² + b².
Sine vs cosine rule
If two angles are known, use sine rule. If the included angle or all three sides are known, use cosine rule.

The Cosine Rule

The cosine rule applies to any triangle. It connects all three sides and one angle. Side c is opposite angle C, which lies between sides a and b.

A B C a b c c² = a² + b² − 2ab cos C C is between sides a and b
Finding a side (SAS — 2 sides + included angle): \[c^2 = a^2 + b^2 - 2ab\cos C\] Finding an angle (SSS — 3 sides known): \[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\]

QCAA Formula Sheet

Which rule to use:
• SAS (2 sides + included angle): cosine rule to find the third side.
• SSS (3 sides): cosine rule to find any angle.
• AAS or ASA (2 angles + 1 side): sine rule.
• SSA (2 sides + non-included angle): sine rule (ambiguous case possible).

Worked Example 1 — Finding a Side (SAS)

Problem: In triangle ABC, a = 8, b = 11, C = 63°. Find side c.

Step 1: Identify: C = 63° is the angle between sides a = 8 and b = 11. Use cosine rule for c.

\[c^2 = a^2 + b^2 - 2ab\cos C\] \[c^2 = 8^2 + 11^2 - 2(8)(11)\cos 63°\] \[c^2 = 64 + 121 - 176 \times 0.4540\] \[c^2 = 185 - 79.90 = 105.10\] \[c = \sqrt{105.10} \approx \mathbf{10.25}\]

Worked Example 2 — Finding All Angles (SSS)

Problem: A triangle has sides a = 7, b = 10, c = 14. Find all angles.

Step 1: Find the largest angle first (opposite longest side c = 14). This is angle C.

\[\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{49 + 100 - 196}{2(7)(10)} = \frac{-47}{140} = -0.3357\] \[C = \cos^{-1}(-0.3357) \approx 109.6°\]

Step 2: Find angle A (opposite a = 7):

\[\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{100 + 196 - 49}{2(10)(14)} = \frac{247}{280} = 0.8821\] \[A \approx \cos^{-1}(0.8821) \approx 28.1°\]

Step 3: B = 180° − 109.6° − 28.1° = 42.3°

Check: 109.6 + 28.1 + 42.3 = 180.0° ✓

Hot Tip: When using SSS, always find the largest angle first (it is opposite the longest side). This angle may be obtuse (greater than 90°). The cosine rule correctly gives obtuse angles because cos&sup8;&sup9; returns values between 0° and 180° — unlike sin&sup8;&sup9; which only returns 0° to 90°.

Where Does the Cosine Rule Come From?

Drop a perpendicular from C to side c. Call the foot D and the height h. Then:

  • AD = b cos A, CD = b sin A
  • BD = c − AD = c − b cos A
  • In right triangle BCD: a² = h² + BD² = (b sin A)² + (c − b cos A)²
  • Expanding: a² = b²sin²A + c² − 2bc cos A + b²cos²A = b² + c² − 2bc cos A

This is the cosine rule. Pythagoras' theorem is the special case when A = 90° (cos 90° = 0).

Step-by-Step: Finding a Side

  1. Identify the angle C and its two adjacent sides a and b.
  2. Substitute into c² = a² + b² − 2ab cos C.
  3. Calculate the right-hand side, then take the square root.

Step-by-Step: Finding an Angle

  1. Identify which angle you want (usually the largest first).
  2. Substitute all three sides into cos C = (a² + b² − c²) / (2ab).
  3. Use cos&sup8;&sup9; to get C. If the result is negative, C is obtuse.
  4. Find remaining angles using sine rule or angle sum.

Guide Example

Problem: Two sides 12 cm and 16 cm, included angle 100°. Find the third side.

\[c^2 = 12^2 + 16^2 - 2(12)(16)\cos 100°\] \[= 144 + 256 - 384 \times (-0.1736)\] \[= 400 + 66.66 = 466.66\] \[c = \sqrt{466.66} \approx 21.60 \text{ cm}\]

Note: cos 100° is negative, so we are adding the last term — the third side is longer than Pythagoras would predict, as expected for an obtuse triangle.

Common Mistake: Writing c² = a² + b² + 2ab cos C (positive instead of negative). Remember: it is always minus 2ab cos C. The minus sign is what makes it work for all angles, not just 90°.

Mastery Practice

See Answers ➔
  1. Fluency Find the missing side using the cosine rule (SAS):
    1. a = 7, b = 10, C = 45°. Find c.
    2. a = 15, b = 20, C = 110°. Find c.
    3. b = 8, c = 12, A = 72°. Find a.
    View Solution
  2. Fluency Find the missing angle using the cosine rule (SSS):
    1. a = 5, b = 8, c = 10. Find angle C.
    2. a = 12, b = 9, c = 15. Find all three angles.
    3. a = 6, b = 6, c = 9. Find the angle between the two equal sides.
    View Solution
  3. Fluency A triangle has sides 14 cm, 18 cm and 22 cm. Find all three angles. View Solution
  4. Understanding Two forces of 50 N and 70 N act at an angle of 58° to each other. Using the triangle of forces (the resultant is the third side of a triangle with the two force magnitudes as the other sides, and the included angle is 180° − 58° = 122°), find the magnitude of the resultant force. View Solution
  5. Understanding A ship travels 45 km on a bearing of 030°, then 60 km on a bearing of 135°. Find the straight-line distance from the starting point to the finishing point.

    Hint: Find the angle between the two legs of the journey using the bearings, then apply the cosine rule.

    View Solution
  6. Understanding A triangular piece of land has sides 85 m, 110 m and 140 m. Find:
    1. All three angles.
    2. The largest angle.
    3. The area of the land using Area = ½ab sin C.
    View Solution
  7. Understanding For each part below, state whether you would use the sine rule or cosine rule and why, then find the required unknown:
    1. B = 55°, a = 14, b = 18. Find angle A.
    2. a = 8, b = 12, C = 47°. Find c.
    3. A = 32°, B = 69°, c = 15. Find a.
    View Solution
  8. Understanding Two forest rangers A and B are 12 km apart. Ranger A sees a fire at an angle of 52° from the line AB. Ranger B sees the fire at an angle of 74° from the line BA. Find:
    1. The distances from each ranger to the fire (use the sine rule).
    2. Verify the result using the cosine rule on the triangle formed.
    View Solution
  9. Problem Solving A triangular park has two sides of 280 m and 350 m. The angle between these sides is 120°. Find:
    1. The length of the third side.
    2. All three angles.
    3. The area of the park.
    4. The perimeter of the park.
    View Solution
  10. Problem Solving Three towns P, Q, R have distances: PQ = 45 km, QR = 72 km, PR = 61 km. A new road is to be built from Q, perpendicular to PR. Find:
    1. Angle QPR (using cosine rule).
    2. The length of the perpendicular from Q to the line PR.
    3. The point on PR where the road meets it (distance from P).
    View Solution