L18 — Sine Rule and Cosine Rule
Key Terms
- Non-right triangle
- Any triangle without a 90° angle — SOH–CAH–TOA does not apply; use the sine or cosine rule instead.
- Sine rule
- a/sin A = b/sin B = c/sin C. Use when you have a matching side–angle pair plus one more piece of information.
- Cosine rule
- a² = b² + c² − 2bc cos A. Use for SAS (two sides + included angle) or SSS (all three sides).
- SAS
- Two sides and the included angle (the angle between the two known sides).
- SSS
- All three sides known; use the cosine rule to find any angle.
- Area formula
- Area = ½ab sin C. Use when two sides and the included angle are known.
When to Use Each Rule
These rules extend trigonometry to any triangle — not just right triangles.
| You know… | Find… | Use… |
|---|---|---|
| AAS or ASA (angle, angle, side) | A side | Sine Rule |
| Two sides + angle opposite one | An angle or side | Sine Rule |
| SAS (two sides + included angle) | Third side | Cosine Rule |
| SSS (all three sides) | An angle | Cosine Rule |
Sine Rule
a / sin A = b / sin B = c / sin C
Flip it to find an angle: sin A / a = sin B / b = sin C / c
Cosine Rule
a² = b² + c² − 2bc cos A
Rearranged to find angle A:
cos A = (b² + c² − a²) / (2bc)
Area of a Triangle
Area = ½ ab sin C
Use when you know two sides and the included angle.
The Sine Rule
In any triangle ABC, the ratio of each side to the sine of its opposite angle is constant. Use it when you have a matching side–angle pair plus one more piece of information.
Worked Example 1 — Find a side using the Sine Rule
In triangle ABC: angle A = 40°, angle B = 75°, side a = 12 cm. Find side b.
Step 1: Angle C = 180 − 40 − 75 = 65°.
Step 2: a/sin A = b/sin B ⇒ 12/sin 40° = b/sin 75°
Step 3: b = 12 × sin 75° / sin 40° = 12 × 0.9659 / 0.6428 ≈ 18.03 cm.
Worked Example 2 — Find an angle using the Sine Rule
In triangle ABC: a = 8, b = 10, angle A = 35°. Find angle B.
sin B / 10 = sin 35° / 8 ⇒ sin B = 10 × sin 35° / 8 = 10 × 0.5736 / 8 = 0.7170
⇒ B = sin²¹(0.7170) ≈ 45.8°.
The Cosine Rule
Use the cosine rule when you do not have a matching side–angle pair. It connects three sides and one angle.
Worked Example 3 — Find a side using the Cosine Rule (SAS)
In triangle ABC: b = 7, c = 9, angle A = 55°. Find side a.
a² = 7² + 9² − 2(7)(9) cos 55° = 49 + 81 − 126 × 0.5736 = 130 − 72.27 = 57.73
a = √57.73 ≈ 7.60 cm.
Worked Example 4 — Find an angle using the Cosine Rule (SSS)
In triangle ABC: a = 5, b = 7, c = 8. Find angle A.
cos A = (b² + c² − a²) / (2bc) = (49 + 64 − 25) / (2 × 7 × 8) = 88/112 = 0.7857
A = cos²¹(0.7857) ≈ 38.2°.
Area of a Non-Right Triangle
Area = ½ab sin C works for any triangle when you know two sides and their included angle.
Worked Example 5 — Area
Triangle with sides 6 m and 9 m, included angle 50°.
Area = ½ × 6 × 9 × sin 50° = 27 × 0.7660 ≈ 20.68 m².
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Use the Sine Rule to find a side. Fluency
Give answers to 2 decimal places.
- (a) A = 30°, B = 70°, a = 10. Find b.
- (b) A = 50°, C = 65°, a = 15. Find c.
- (c) B = 45°, C = 80°, b = 8. Find c.
- (d) A = 25°, B = 110°, b = 20. Find a.
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Use the Sine Rule to find an angle. Fluency
Give angles to 1 decimal place.
- (a) a = 6, b = 8, A = 40°. Find B.
- (b) a = 10, c = 12, A = 55°. Find C.
- (c) b = 15, c = 10, B = 70°. Find C.
- (d) a = 5, b = 5, A = 50°. Find B (and explain your answer).
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Use the Cosine Rule to find a side. Fluency
Give answers to 2 decimal places.
- (a) b = 5, c = 7, A = 60°. Find a.
- (b) a = 9, c = 12, B = 45°. Find b.
- (c) a = 4, b = 6, C = 100°. Find c.
- (d) b = 8, c = 8, A = 90°. Find a (and verify with Pythagoras).
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Use the Cosine Rule to find an angle. Fluency
Give angles to 1 decimal place.
- (a) a = 7, b = 8, c = 10. Find A.
- (b) a = 5, b = 6, c = 7. Find B (the middle-sized angle).
- (c) a = 3, b = 4, c = 5. Find C (the largest angle).
- (d) a = 12, b = 12, c = 12. Find any angle.
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Read from the triangle diagram. Understanding
The diagram shows triangle PQR with the measurements marked.
- (a) Find angle R.
- (b) Which rule should you use to find QR? Explain why.
- (c) Find the length QR (opposite to angle P = 48°) to 2 decimal places.
- (d) Find the length PR (opposite to angle Q = 62°) to 2 decimal places.
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Area of a triangle. Understanding
- (a) Sides 8 cm and 11 cm, included angle 40°. Find the area.
- (b) Sides 5 m and 12 m, included angle 90°. Find the area using ½ab sinC, then verify with ½bh.
- (c) A triangle has area 30 cm² and two sides of 10 cm and 8 cm. Find the included angle.
- (d) An equilateral triangle has side 6 cm. Find the exact area using ½ab sinC.
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Choose the correct rule. Understanding
For each triangle, state whether to use the Sine Rule or Cosine Rule and why.
- (a) Two angles and one side are known.
- (b) Two sides and the included angle are known.
- (c) Two sides and the angle opposite one of them are known.
- (d) All three sides are known and you want an angle.
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Surveying problem. Understanding
A surveyor measures a triangular block of land. One side is 120 m. The angles at each end of that side are 58° and 67°.
- (a) Find the third angle of the triangle.
- (b) Find the two unknown sides.
- (c) Find the area of the block of land.
- (d) The block is to be fenced along its perimeter. Find the total fencing required.
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Two boats and a lighthouse. Problem Solving
Boat A is 5.2 km from a lighthouse. Boat B is 3.8 km from the same lighthouse. The angle between the lines of sight from the lighthouse to the two boats is 72°.
- (a) Draw a labelled diagram.
- (b) Identify which rule you need and explain why.
- (c) Find the distance between Boat A and Boat B.
- (d) Find the angle at Boat A (the angle between the line to the lighthouse and the line to Boat B).
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Navigation: three towns. Problem Solving
Town B is 45 km from Town A on a bearing of 035°. Town C is 60 km from Town A on a bearing of 110°.
- (a) Draw a diagram showing A, B, and C with the bearings.
- (b) Find the angle BAC (the angle at A between the directions to B and C).
- (c) Find the straight-line distance from B to C.
- (d) Find the bearing of C from B.