Choosing the right tool
To solve a triangle that has no right angle (an oblique triangle), choose between the sine law and the cosine law based on what’s given.
| Given | First step | Then |
|---|---|---|
| AAS or ASA | sine law | angle sum, sine law |
| SSA | sine law (check ambiguous case) | angle sum, sine law |
| SAS | cosine law for the third side | sine law or cosine law for an angle |
| SSS | cosine law for the largest angle | sine law, angle sum |
Remember: the three angles always add to 180°.
Example: SAS
b = 9, c = 14, A = 48°
- Cosine law: a² = 81 + 196 − 252 cos 48° ≈ 277 − 168.6 = 108.4 → a ≈ 10.41
- B is opposite the shorter of b and c, so it’s acute. Sine law: sin B = 9 sin 48°/10.41 ≈ 0.6425 → B ≈ 40.0°
- C = 180 − 48 − 40.0 = 92.0°
Example: SSS
a = 6, b = 8, c = 11
- Largest angle C: cos C = (36 + 64 − 121)/96 ≈ −0.219 → C ≈ 102.6°
- Sine law: sin A = 6 sin 102.6°/11 ≈ 0.532 → A ≈ 32.2°
- B = 180 − 102.6 − 32.2 = 45.2°
Area of any triangle
Area = ½ab sin C (two sides and the included angle)
From the SAS example: ½(9)(14) sin 48° ≈ 46.8 square units.
When you know all three sides, Heron’s formula also works: A = √[s(s − a)(s − b)(s − c)], where s is half the perimeter.
Bearings
A bearing is a direction measured clockwise from north (000° to 360°).
Example: A plane flies 200 km on a bearing of 060°, then 150 km on a bearing of 150°. How far is it from the start?
- The turn between the legs: 150° − 60° = 90° → the angle inside the triangle is 90°
- d = √(200² + 150²) = 250 km
(If the angle isn’t 90°, use the cosine law.)
Real-world applications
- Surveying land with inaccessible sides
- Navigation for ships and planes
- Engineering trusses and roof frames
- Finding distances across lakes or canyons
Worked examples
Example 1: A = 35°, B = 75°, c = 20. Find a. C = 70° → a = 20 sin 35°/sin 70° ≈ 12.2.
Example 2: Area with a = 7, b = 10, C = 30°: ½(7)(10)(0.5) = 17.5.
Example 3: Which law first for a = 5, b = 7, c = 9? Cosine law (SSS).
Common mistakes
- Using the sine law when you don’t have a matching pair.
- Rounding intermediate values too much.
- Forgetting the ambiguous case with SSA.
Try it yourself
- Which law for A = 40°, b = 12, c = 15?
- Solve: A = 40°, b = 12, c = 15 (find a).
- Area of the triangle in question 2.
- A ship sails 30 km north, then 40 km on a bearing of 120°. Angle at the turn inside the triangle?
Answers: 1) cosine law (SAS) 2) a ≈ 9.65 3) ≈ 57.9 square units 4) 60°
Quick recap
- Pick the sine law or cosine law based on the given information.
- Area formula: ½ab sin C.
- Bearings are measured clockwise from north.