Library / Geometry, coordinates, surveying and trigonometry / GEO-46
Geometry, coordinates, surveying and trigonometry
Period of a transformed sine or cosine
Fundamental period of y = A sin(bx + c) + d, or the cosine form, with the argument in radians.
- Ref
- GEO-46
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- T [x units]
For y = A sin(bx + c) + d, the sine completes one cycle when its argument goes through 2π, so the x-distance of one cycle is 2π/|b|. The amplitude A, the shift c and the offset d do not change it. The same period holds for the cosine. With x in degrees the period is 360°/|b| instead. The coefficient b cannot be zero.
Fundamental period of y = A sin(bx + c) + d, or the cosine form, with the argument in radians. A, c and d do not change it. b cannot be zero.
Calculator
- Frequency in cycles per unit x, |b|/(2π)0.6366 1/x unit
- Half period0.7854 x units
- Period if x is in degrees, 360°/|b|90 x units
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.
- y = sin(−4x) repeats every 1.571 units, 0.6366 cycles per unit.
- Doubling b halves the period, so a wave twice as fast repeats twice as often.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (x units) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Coefficient of x in the argument | −10 to −0.5 1/unit | 0.6283 | 12.57 | 11.94 | +0.19 1/unit | +4.99 % |
Sweep of Coefficient of x in the argument
| Coefficient of x in the argument (1/unit) | Period (x units) |
|---|---|
| −10 | 0.6283 |
| −8.813 | 0.713 |
| −7.625 | 0.824 |
| −6.438 | 0.976 |
| −5.25 | 1.197 |
| −4.063 | 1.547 |
| −2.875 | 2.185 |
| −1.688 | 3.723 |
| −0.5 | 12.57 |
Sweep one variable
Variables and how each one contributes
Period
What the equation gives you. Everything else feeds it.
Coefficient of x in the argument
The coefficient of x inside the sine or cosine. The sign does not change the period; only its size matters. A larger size of b gives a shorter period, in inverse proportion. Raising it by 0.5 (to −3.5) moves the result by +14.3 %, from 1.571 to 1.795 x units.
Worked example with the values above
- Size of the coefficient: |b| = |−4| = 4.
- One cycle needs the argument to change by 2π = 6.2832.
- T = 6.2832 / 4 = 1.5708 x units.
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