A sinusoid is any curve shaped like a sine wave: tides, daylight hours, alternating current, sound and seasonal temperatures all follow one. Four numbers describe it completely — amplitude, period, phase shift and midline. This calculator reads them off an equation, lists the five key points of a cycle and draws two periods, or works the other way and builds the equation from a maximum, a minimum and the period.
How to use the sinusoidal function calculator
- Choose Find amplitude, period, phase shift and midline or Write the equation from a maximum, minimum and period.
- To analyze, pick sine or cosine and the equation form —
A·f(Bx − C) + DorA·f(B(x − C)) + D— then enter A, B, C and D. Expressions such aspi/2work, and x can be in radians or degrees. - To build, enter the maximum value, the minimum value, the period and an x where a maximum occurs.
- Read the equation and its features on the tape, the key points in the table, and the graph with its dashed midline.
Sinusoid formulas
For y = A sin(B(x − h)) + D or the matching cosine:
| Feature | Formula |
|---|---|
| Amplitude | |A| |
| Period | 2π/|B| (or 360°/|B|) |
| Frequency | |B|/2π cycles per unit |
| Phase shift | h (right if positive) |
| Midline | y = D |
| Maximum, minimum | D + |A|, D − |A| |
In the form A sin(Bx − C) + D, the phase shift is h = C/B, not C — the single most common slip with these problems.
Worked examples
Analyze y = 3 sin(2x − π/2) + 1.
Amplitude |3| = 3; period 2π/2 = π; phase shift (π/2) ÷ 2 = π/4 to the right; midline y = 1; maximum 4 and minimum −2.
Key points of one cycle, starting at x = π/4: (π/4, 1) on the midline, (π/2, 4) at the maximum, (3π/4, 1), (π, −2) at the minimum and (5π/4, 1). As a cosine, the same curve is y = 3 cos(2(x − π/2)) + 1.
Build a model for monthly average temperature that peaks at 18 °C in month 7, bottoms out at 4 °C, and repeats every 12 months.
Amplitude (18 − 4)/2 = 7; midline (18 + 4)/2 = 11; B = 2π/12 = π/6. A cosine starts at its peak, so shift it to x = 7: y = 7 cos((π/6)(x − 7)) + 11. The sine version shifts a quarter period (3 months) earlier: y = 7 sin((π/6)(x − 4)) + 11.
Reading the five key points
Each cycle of a sinusoid can be sketched from five evenly spaced points, a quarter period apart:
- Sine (with positive A) goes midline → maximum → midline → minimum → midline.
- Cosine (with positive A) goes maximum → midline → minimum → midline → maximum.
- A negative A swaps maximum and minimum.
Plot those points, join them with a smooth wave, and repeat the pattern to the left and right.
Where sinusoids appear
| Situation | Typical period |
|---|---|
| Household AC voltage in the US | 1/60 second |
| Ocean tides (semidiurnal) | about 12.4 hours |
| Daylight hours over a year | 365 days |
| Middle C on a piano | about 1/262 second |
Real data is rarely a perfect sinusoid, but fitting amplitude, midline and period from the highest and lowest values gives a model that is often good enough for planning and prediction.
Related tools
To explore all six trig functions with sliders for A, B, C and D, use the trigonometric function graphs. The unit circle calculator explains where the wave shape comes from, and the trigonometric identities checker verifies conversions such as sin u = cos(u − π/2).
Frequently asked questions
How do I find the phase shift of y = A sin(Bx − C) + D?
Factor B out of the argument: Bx − C = B(x − C/B). The phase shift is C/B, to the right when positive. For y = 3 sin(2x − π/2) + 1, the shift is (π/2)/2 = π/4 to the right.
What is the period of a sinusoid?
The length of one full cycle: 2π/|B| in radians or 360°/|B| in degrees. Larger B squeezes more cycles into the same interval.
What do A and D control?
|A| is the amplitude, the distance from the midline to a peak. D is the vertical shift, so the midline is y = D, the maximum is D + |A| and the minimum is D − |A|. A negative A flips the curve upside down.
How do I write an equation from a graph or data?
Read the maximum and minimum: amplitude = (max − min)/2 and midline = (max + min)/2. Measure the period to get B = 2π/period. A cosine starts at a maximum, so shift it to the x-value of any maximum.
Can a sine function be written as a cosine?
Always. sin u = cos(u − π/2), so a sine graph is a cosine graph shifted a quarter period to the right. The calculator gives the equivalent form of every equation.