Library / Geometry, coordinates, surveying and trigonometry / GEO-48
Geometry, coordinates, surveying and trigonometry
Cosine equation solutions in one revolution
The smaller of the two angles between 0 and 2π with cos x = c, where c lies between −1 and 1.
- Ref
- GEO-48
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- x1 [rad]
The equation cos x = c has two solutions in one revolution, x₁ = arccos c, which lies between 0 and π, and x₂ = 2π − arccos c. The calculator shows the smaller one, arccos c, and the larger is in the derived rows. For c = 1 only x = 0 remains, and for c = −1 only x = π.
The smaller of the two angles between 0 and 2π with cos x = c, where c lies between −1 and 1. The other, 2π − arccos c, is in the derived rows. At c = ±1 only one solution remains.
Calculator
- Larger solution, 2π − arccos c4.189 rad
- Smaller solution in degrees120 °
- Gap between the two solutions2.094 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.
- cos x = −0.5 has solutions 2.0944 and 4.1888 rad in one revolution, mirror images about π.
- At c = 1 only 0 remains, and at c = −1 only π, because the two solutions merge.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| c Value of cos x | −1 to 1 | 0 | 3.142 | 3.142 | +0.04 | −2.18 % |
Sweep of Value of cos x
| Value of cos x | Smaller solution (rad) |
|---|---|
| −1 | 3.142 |
| −0.75 | 2.419 |
| −0.5 | 2.094 |
| −0.25 | 1.823 |
| 0 | 1.571 |
| 0.25 | 1.318 |
| 0.5 | 1.047 |
| 0.75 | 0.7227 |
| 1 | 0 |
Sweep one variable
Variables and how each one contributes
Smaller solution
What the equation gives you. Everything else feeds it.
Value of cos x
The value of cos x, a plain number from −1 to 1. Lowers the smaller solution as c rises, since arccos falls when c increases. Raising it by 0.05 (to −0.45) moves the result by −2.71 %, from 2.094 to 2.038 rad.
Worked example with the values above
- Principal value: arccos(−0.5) = 2.0944 rad.
- The other solution is 2π − 2.0944 = 4.1888 rad.
- Smaller solution: x₁ = 2.0944 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