Library / Geometry, coordinates, surveying and trigonometry / GEO-47
Geometry, coordinates, surveying and trigonometry
Sine equation solutions in one revolution
The smaller of the two angles between 0 and 2π with sin x = s, where s lies between −1 and 1.
- Ref
- GEO-47
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- x1 [rad]
The equation sin x = s has two solutions in one revolution, x₁ = arcsin s and x₂ = π − arcsin s, each taken modulo 2π so that they lie between 0 and 2π. The calculator shows the smaller one; the larger is in the derived rows. At s = ±1 the two coincide, giving one solution. The calculator’s own arcsin gives only the principal value.
The smaller of the two angles between 0 and 2π with sin x = s, where s lies between −1 and 1. The other is in the derived rows. At s = ±1 the two coincide.
Calculator
- Larger solution, in radians5.76 rad
- Smaller solution in degrees210 °
- Principal value arcsin s−0.5236 rad
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.
- sin x = −0.5 has solutions 3.6652 and 5.7596 rad in one revolution (210° and its partner).
- At s = 1 there is only one solution, π/2, and at s = 0 the solutions are 0 and π.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Value of sin x | −1 to 1 | 0 | 4.712 | 4.712 | +0.04 | −1.24 % |
Sweep of Value of sin x
| Value of sin x | Smaller solution (rad) |
|---|---|
| −1 | 4.712 |
| −0.75 | 3.99 |
| −0.5 | 3.665 |
| −0.25 | 3.394 |
| 0 | 0 |
| 0.25 | 0.2527 |
| 0.5 | 0.5236 |
| 0.75 | 0.8481 |
| 1 | 1.571 |
Sweep one variable
Variables and how each one contributes
Smaller solution
What the equation gives you. Everything else feeds it.
Value of sin x
The value of sin x, a plain number from −1 to 1. Moves both solutions, in opposite directions when s is between −1 and 1. Raising it by 0.05 (to −0.45) moves the result by −1.55 %, from 3.665 to 3.608 rad.
Worked example with the values above
- Principal value: arcsin(−0.5) = −0.5236 rad.
- The two solutions are −0.5236 and π − −0.5236 = 3.6652, each reduced into 0 to 2π.
- Smaller solution: x₁ = 3.6652 rad.
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