The Sine Rule
Key Terms
- The sine rule relates sides and opposite angles in any triangle
- Use when given: AAS (two angles + one side), ASA (two angles + included side), or SSA (two sides + non-included angle)
- The ambiguous case (SSA) occurs when two different triangles can satisfy the given information
- The area formula using sine: Area = ½ab sin C, where C is the included angle
- Always check: can the angle be obtuse? sin θ = sin(180° − θ)
a/sin A = b/sin B = c/sin C
Equivalently: sin A/a = sin B/b = sin C/c
Area of a Triangle (Sine Formula)
Area = ½ab sin C = ½ac sin B = ½bc sin A
When to use the Sine Rule:
• AAS: two angles and any side known
• ASA: two angles and the included side known
• SSA: two sides and a non-included angle known (check ambiguous case!)
Step 1: Identify what's given: A = 40°, B = 75°, a = 8. Find b.
Step 2: Apply the sine rule: b/sin B = a/sin A
b/sin 75° = 8/sin 40°
b = 8 × sin 75° / sin 40°
b = 8 × 0.9659 / 0.6428
b ≈ 12.0 cm
Step 1: Apply the sine rule: sin B / b = sin A / a
sin B = 10 × sin 35° / 7 = 10 × 0.5736 / 7 ≈ 0.8194
Step 2: Find angle B:
B1 = sin−1(0.8194) ≈ 55.0°
B2 = 180° − 55.0° = 125.0°
Step 3: Check validity:
• If B = 55.0°: C = 180 − 35 − 55 = 90° > 0° ✓
• If B = 125.0°: C = 180 − 35 − 125 = 20° > 0° ✓
Both triangles are valid — this is the ambiguous case. B ≈ 55.0° or B ≈ 125.0°.
Beyond Right-Angled Triangles
In earlier studies, trigonometry was restricted to right-angled triangles, where sin, cos, and tan relate sides to the right angle. But most real triangles — those used in navigation, surveying, and engineering — are not right-angled. The Sine Rule extends trigonometry to any triangle by connecting each side to the sine of its opposite angle.
Label a triangle with vertices A, B, C and opposite sides a, b, c respectively. The sine rule states:
a/sin A = b/sin B = c/sin C
This elegant relationship holds for all triangles, acute or obtuse. The proof follows from drawing the perpendicular height h from one vertex: for example, h = b sin A = a sin B, which rearranges directly to the sine rule.
When to Use the Sine Rule
The sine rule works whenever you can pair a side with its opposite angle. Specifically, you need at least one known side-angle pair, plus one additional piece of information:
AAS (Angle-Angle-Side): Two angles and any side. Since angles sum to 180°, you can find the third angle immediately, then apply the sine rule to find any side.
ASA (Angle-Side-Angle): Two angles and the included side. Again, find the third angle first, then use the sine rule. This always gives a unique triangle.
SSA (Side-Side-Angle): Two sides and a non-included angle. This is the ambiguous case — proceed carefully, as there may be two valid triangles, one valid triangle, or no valid triangle at all.
The Ambiguous Case Explained
When given two sides (a, b) and an angle (A) opposite to the shorter of the two given sides, it is geometrically possible that two different triangles fit the description. Imagine fixing side b and angle A, then swinging side a like a compass arc — it may intersect the base line in two places (two triangles), one place (one triangle), or not at all (no triangle).
The rule of thumb: when A is acute and a < b, check whether 180° − B1 still leaves a positive angle for C. If yes, both triangles are valid. Always check both solutions in an exam.
When A is obtuse, the obtuse possibility for B would make A + B > 180°, which is impossible, so there is at most one solution.
Area Using the Sine Formula
The formula Area = ½ab sin C is derived from the standard area = ½ × base × height. Drawing height h from vertex C to base c gives h = a sin B (or equivalently a sin B), and from h = b sin A. Substituting: Area = ½ × c × h. The most useful form for the sine rule context is:
Area = ½ab sin C
where C is the included angle between sides a and b. This works for any triangle without needing the perpendicular height.
Bearings and Real-World Applications
Navigation problems often describe distances and bearings (angles measured clockwise from north). When two ships travel on known bearings from a port, the angle between their paths and the triangle formed by their positions can be solved using the sine rule. The key step is always to extract the interior angles of the triangle from the bearing information — drawing a clear diagram is essential.
Mastery Practice
- In triangle ABC, A = 50°, B = 70°, and a = 9 cm. Find the length of side b.
- In triangle PQR, P = 35°, Q = 80°, and q = 14 m. Find the length of side p.
- In triangle ABC, a = 6, b = 9, and A = 30°. Find angle B (include all possible solutions).
- Find the area of triangle ABC where a = 8 cm, b = 11 cm, and the included angle C = 55°.
- In triangle ABC, B = 62°, C = 48°, and b = 15 m. Find side c and the area of the triangle.
- In triangle XYZ, x = 10, y = 6, and X = 120°. Explain why there is only one possible triangle, and find angle Y.
- Two points A and B are on opposite sides of a river. From a point C, 80 m from A, angle ACB = 62° and angle CAB = 74°. Find the distance AB.
- In triangle ABC, a = 5, b = 7, and A = 45°. Determine whether the ambiguous case applies and find all possible triangles. For each valid triangle, state the area.
- A ship leaves port P and travels 12 km on a bearing of 040° to point A. A second ship travels from P on a bearing of 110°. The angle PAB (at point A, looking from P toward B) is 35°. Find the distance AB.
- In triangle ABC, the area is 24 cm², side a = 8 cm, side b = 7 cm. Use the area formula to find angle C. Then use the sine rule to find side c. Give answers to 1 decimal place.