Beyond right triangles
SOH CAH TOA works only in right triangles. For an oblique triangle (no right angle), we use the sine law or the cosine law.
The sine law
In any triangle ABC, with side a opposite angle A, and so on:
a / sin A = b / sin B = c / sin C
or, flipped (handy for finding angles):
sin A / a = sin B / b = sin C / c
Each side divided by the sine of its opposite angle gives the same value.
When to use the sine law
You need a "matching pair": a side and its opposite angle, plus one more piece.
| Given | Use |
|---|---|
| AAS (two angles, a non-included side) | sine law |
| ASA (two angles, included side) | find the third angle, then sine law |
| SSA (two sides, a non-included angle) | sine law, but watch the ambiguous case |
| SAS or SSS | cosine law instead |
Finding a side
Example: A = 40°, B = 65°, a = 10. Find b.
- b/sin 65° = 10/sin 40°
- b = 10 sin 65° / sin 40° ≈ 10(0.9063)/0.6428 ≈ 14.1
Finding an angle
Example: a = 12, b = 9, A = 70°. Find B.
- sin B / 9 = sin 70° / 12
- sin B = 9 sin 70° / 12 ≈ 0.7048
- B ≈ 44.8°
The ambiguous case (SSA)
Because sin θ = sin(180° − θ), an angle found with sin⁻¹ might have a second possibility.
Example: A = 30°, a = 6, b = 10. Find B.
- sin B = 10 sin 30° / 6 ≈ 0.833 → B ≈ 56.4° or B ≈ 123.6°
- Both work (30 + 123.6 < 180), so two triangles are possible.
| Situation (A acute) | Triangles |
|---|---|
| a < b sin A | 0 |
| a = b sin A | 1 (right triangle) |
| b sin A < a < b | 2 |
| a ≥ b | 1 |
Real-world example
Two ranger stations 12 km apart spot a fire. From station P, the fire is at an angle of 52° from the line PQ; from station Q, 61°. How far is the fire from P?
- Third angle: 180 − 52 − 61 = 67°
- d/sin 61° = 12/sin 67° → d ≈ 12(0.8746)/0.9205 ≈ 11.4 km
Worked examples
Example 1: A = 50°, C = 70°, c = 15. Find a. a = 15 sin 50°/sin 70° ≈ 12.2.
Example 2: B = 35°, b = 8, c = 12. Find C. sin C = 12 sin 35°/8 ≈ 0.860 → C ≈ 59.4° or 120.6° (two triangles).
Example 3: Why can’t the sine law solve SSS? There’s no known angle to start a matching pair.
Common mistakes
- Pairing a side with the wrong angle.
- Missing the second solution in the ambiguous case.
- Calculator in radian mode.
Try it yourself
- A = 45°, B = 60°, a = 8. Find b.
- a = 10, A = 80°, b = 7. Find B.
- A = 25°, B = 105°, c = 20. Find a.
- How many triangles: A = 40°, a = 5, b = 12?
Answers: 1) ≈ 9.8 2) ≈ 43.6° 3) ≈ 11.0 4) none (12 sin 40° ≈ 7.7 > 5)
Quick recap
- Sine law: a/sin A = b/sin B = c/sin C.
- Use it for AAS, ASA and SSA.
- SSA can be the ambiguous case: check for 0, 1 or 2 triangles.