Sine Rule and Cosine Rule
Master the sine rule, cosine rule, and the ambiguous case for solving non-right triangles, along with the area formula using trigonometry in HSC Advanced Mathematics.
The Sine Rule
The sine rule relates the sides and angles of any triangle. In triangle ABC with sides a, b, c opposite angles A, B, C respectively:
a/sin A = b/sin B = c/sin C
Use the sine rule when you know an angle-side opposite pair and one other piece of information.
The sine rule is used when you have: (1) two angles and one side (AAS), or (2) two sides and a non-included angle (SSA) — but beware of the ambiguous case.
Example: In triangle ABC, A = 40°, B = 70°, a = 8. Find b.
8/sin 40° = b/sin 70°
b = 8 × sin 70° / sin 40° = 8 × 0.9397 / 0.6428 ≈ 11.7
The Cosine Rule
The cosine rule is a generalisation of Pythagoras' theorem for any triangle. It is used when the sine rule cannot be applied directly.
c2 = a2 + b2 − 2ab cos C
To find an angle: cos C = a2 + b2 − c22ab
Use the cosine rule when you have: (1) two sides and the included angle (SAS) — to find the third side, or (2) all three sides (SSS) — to find an angle.
Example: In triangle PQR, p = 7, q = 10, R = 60°. Find r.
r2 = 72 + 102 − 2(7)(10)cos 60° = 49 + 100 − 140 × 0.5 = 79.
r = √79 ≈ 8.89
The Ambiguous Case and Area Formula
The ambiguous case arises when using the sine rule with two sides and a non-included angle (SSA). Since sin θ = sin(180° − θ), there may be 0, 1, or 2 valid triangles.
Two solutions: If B is acute and a > b sin A (but a < b), there are two possible triangles.
One solution: If the angle is obtuse, or a ≥ b.
No solution: If a < b sin A, the triangle is impossible.
The area of a triangle using trigonometry is:
Area = 1/2ab sin C
Where a and b are two sides and C is the included angle.
Example: Find the area of a triangle with sides 6 and 9 and included angle 50°.
Area = ½ × 6 × 9 × sin 50° = 27 × 0.766 ≈ 20.7 square units.
Key Vocabulary
Sine Rule
Relates sides and opposite angles: a/sin A = b/sin B = c/sin C.
Cosine Rule
Generalises Pythagoras: c2 = a2 + b2 − 2ab cos C.
Ambiguous Case
When SSA data can produce 0, 1, or 2 valid triangles due to the sine function's symmetry.
Included Angle
The angle between two known sides, required for the cosine rule and area formula.
Worked Examples
In triangle ABC, a = 12, b = 8, C = 45°. Find c using the cosine rule.
Step 1: c2 = 122 + 82 − 2(12)(8)cos 45°
Step 2: c2 = 144 + 64 − 192 × 0.7071 = 208 − 135.76 = 72.24
Answer: c = √72.24 ≈ 8.50
In triangle XYZ, x = 5, y = 7, z = 9. Find angle Z.
Step 1: cos Z = (x2 + y2 − z2) / (2xy) = (25 + 49 − 81) / (2 × 5 × 7) = −7/70 = −0.1
Answer: Z = cos−1(−0.1) ≈ 95.7°
Investigate the ambiguous case: A = 30°, a = 5, b = 8. How many triangles are possible?
Step 1: Check b sin A = 8 × sin 30° = 8 × 0.5 = 4.
Step 2: Since a = 5 > 4 = b sin A and a < b, there are two possible triangles.
Step 3: sin B = 8 sin 30° / 5 = 0.8. So B = 53.1° or B = 126.9°.
Answer: Two triangles: one with B ≈ 53.1° and one with B ≈ 126.9°.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
When should you use the cosine rule rather than the sine rule?
Question 2
In triangle ABC, A = 35°, C = 80°, a = 10. Find c (to 1 d.p.).
Question 3
Find the area of a triangle with sides 10 cm and 14 cm and an included angle of 30°.
Question 4
In a triangle with sides 5, 6, 7, find the largest angle (to nearest degree).
Question 5
In the ambiguous case with A = 40°, a = 5, b = 7, how many triangles are possible?
Key Concepts Summary
- ● The sine rule a/sin A = b/sin B = c/sin C is used for AAS and SSA cases.
- ● The cosine rule c2 = a2 + b2 − 2ab cos C is used for SAS and SSS cases.
- ● The ambiguous case (SSA) may give 0, 1, or 2 triangles — always check.
- ● The area formula A = ½ab sin C uses two sides and the included angle.
- ● The largest angle is always opposite the longest side.