Library / Geometry, coordinates, surveying and trigonometry / GEO-50
Geometry, coordinates, surveying and trigonometry
Second angle in the ambiguous sine rule case
Smaller possible value of the angle B from the sine rule when the known angle A is opposite side a and side b is given.
- Ref
- GEO-50
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- B [rad]
In the sine rule sin B / b = sin A / a, the equation sin B = b sin A / a has two solutions between 0 and π: B and π − B. Both are triangles when the second leaves room for the known angle, π − B < π − A, which means B > A. If b sin A / a exceeds 1 there is no triangle. This entry shows the smaller candidate; the other is in the derived rows.
Smaller possible value of the angle B from the sine rule when the known angle A is opposite side a and side b is given. The supplementary angle π − B, which may give a second triangle, is in the derived rows. If b sin A / a exceeds 1 there is no triangle.
Calculator
- Other candidate, π − B2.356 rad
- Second triangle exists (1 = yes, 0 = no)1
- Smaller candidate in degrees45 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With a = 7 cm, b = 9.899 cm and A = 30°, angle B can be 45° or 135°, giving two triangles.
- If b sin A / a is exactly 1 there is a single right-angled triangle, and above 1 none; for A = 45° with a = 1 and b = 3 no triangle exists.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Side opposite angle A | 5 to 30 cm | 0.1657 | 1.429 | 1.263 | +10% | −11.1 % |
| A Known angle A | 0.1 to 0.75 rad | 0.1417 | 1.302 | 1.16 | +10% | +11.9 % |
| b Side opposite angle B | 2 to 13 cm | 0.1433 | 1.191 | 1.047 | +10% | +13.5 % |
Sweep of Side opposite angle B
| Side opposite angle B (cm) | Smaller candidate for angle B (rad) |
|---|---|
| 2 | 0.1433 |
| 3.375 | 0.2435 |
| 4.75 | 0.3462 |
| 6.125 | 0.4528 |
| 7.5 | 0.5654 |
| 8.875 | 0.6866 |
| 10.25 | 0.8215 |
| 11.63 | 0.9797 |
| 13 | 1.191 |
Sweep one variable
Variables and how each one contributes
Smaller candidate for angle B
What the equation gives you. Everything else feeds it.
Side opposite angle A
The length of the side opposite the known angle A, in cm. A longer side a lowers sin B and so lowers the smaller candidate for B. Raising it by 10 % (to 7.7) moves the result by −11.1 %, from 0.7854 to 0.6982 rad.
Side opposite angle B
The length of the side opposite the unknown angle B, in cm. A longer side b raises sin B, and so raises the smaller candidate for B; if it passes a/sin A there is no triangle. Raising it by 10 % (to 10.89) moves the result by +13.5 %, from 0.7854 to 0.8912 rad.
Known angle A
The known angle opposite side a, in radians, between 0 and π. A larger angle raises sin B up to a right angle, then lowers it again. Raising it by 0.05 (to 0.5736) moves the result by +11.4 %, from 0.7854 to 0.8748 rad.
Worked example with the values above
- Sine of B: sin B = b sin A / a = 9.899 × 0.5 / 7 = 0.70711.
- Candidates: B = arcsin(0.70711) = 0.7854 rad and π − B = 2.3562 rad.
- Smaller candidate: 0.7854 rad, and the second one also gives a valid triangle.
Assumptions and limits
- Angles in radians, and a principal value of the inverse function.
- For the sine rule, the sides and angle are consistent with a triangle that exists.
Common mistakes
- Giving only the calculator’s principal solution and missing the second one in the interval.
- Forgetting to add whole multiples of π for tangent or 2π for sine and cosine in a general solution.
- Using a period for the argument bx when x is measured in degrees.
Statistics and uncertainty
Each answer is exact but may sit on a boundary: near ±1 an inverse sine or cosine is very sensitive to rounding, and in the ambiguous case a small change in the data can add or remove a second triangle.
Where it is used
- Solving periodic problems: tides, alternating currents, vibration, rotating machinery.
- Solving triangles from two sides and a non-included angle.
Maths you need
- Inverse trigonometric functions: arcsin, arccos and arctan
- Period: the length of one full cycle
- Sine rule: a/sin A = b/sin B