Find the Period of a Sine Function: Formula and Examples

The period of a sine function is the horizontal length of one complete cycle — the distance after which the wave repeats itself. For the standard form y = A sin(Bx + C) + D, you find the period of a sine function with the formula T = 2π / |B|. This means the period depends only on the coefficient B that multiplies the variable x; the amplitude A, phase shift C, and vertical shift D do not affect it.

That single formula answers most period questions you’ll meet in trigonometry and precalculus. The rest of this article shows exactly how to apply it and read the period straight off a graph when no equation is given.

Sine Period Formula

The core relationship is T = 2π / |B|, where T is the period. The vertical bars indicate absolute value, so a negative B produces the same period as its positive counterpart.

Here is what each part of y = A sin(Bx + C) + D controls:

• A — amplitude: the height from the midline to a peak. It stretches the wave vertically but never changes the period.
• B — frequency coefficient: the number that multiplies x. It compresses or stretches the wave horizontally and is the only value in the period formula.
• C — phase shift: slides the whole graph left or right. The shift is −C/B, and it moves the wave without resizing a cycle.
• D — vertical shift: raises or lowers the midline. It changes vertical position only.

Because A, C, and D act vertically or shift the curve without changing its horizontal width, the period is set entirely by B. For the basic untransformed curve y = sin(x), B = 1, so T = 2π/1 = 2π ≈ 6.283. That 2π length is the natural full cycle of the sine wave, a fact covered in most Khan Academy trigonometry units.

A useful intuition: larger |B| means more cycles squeezed into the same horizontal space, so each individual period becomes shorter.

Common traps to avoid

The most frequent mistake is reading the wrong number for B when the argument isn’t in simple Bx form. For instance, in y = sin(2x + π), the B is 2, not the π inside the parentheses — that π is part of the phase shift C, not the frequency.

Another trap: confusing B with the period itself. A coefficient of 2 does not mean a period of 2; it means T = 2π/2 = π.

A Worked Example

The quickest way to get comfortable is to run a B value through the formula.

Find the period of y = 2 sin(0.5x − π) − 4.

The coefficient of x is 0.5, so B = 0.5 — ignore the −π and the −4. T = 2π / 0.5 = 4π ≈ 12.566.

Notice how the amplitude 2 and vertical shift −4 play no role. The fractional coefficient stretches the wave, producing a period twice as long as the standard 2π.

The same logic covers other cases. If B is a whole number like 4, then T = 2π/4 = π/2, a shorter cycle. If B is negative, say −3, the absolute value gives T = 2π/3. The negative sign reflects the graph across the y-axis, but reflection doesn’t change how long one cycle lasts. And if the equation is y = sin(x/2), writing it as (1/2)x makes B = 1/2, so T = 2π / (1/2) = 4π.

Putting these together shows a clear pattern: multiplying B by a number divides the period by that same number. Course notes on periodic functions, such as MIT OpenCourseWare materials on trigonometric graphs, reinforce this inverse relationship between frequency and period.

Period from a Graph

When you’re handed a picture instead of an equation, you can read the period directly — no formula rearrangement needed. The period is simply the horizontal distance between any two matching points where the pattern repeats.

Step-by-step from a graph:

1. Pick a clear landmark. Choose a point that’s easy to identify, such as a peak (maximum) or a trough (minimum).
2. Find the next identical landmark. Locate the very next peak of the same height, or the next trough — not a half-cycle feature like a zero crossing.
3. Measure horizontally. Read the x-values of the two landmarks and subtract them: period = x₂ − x₁.
4. Sanity-check. Confirm the distance between several consecutive peaks stays constant, which it must for a true periodic sine wave.

Peaks, troughs, and half-cycles

A common error is measuring between a peak and the next trough. That span is only half a period. The full period runs peak-to-peak or trough-to-trough.

For example, if a graph shows a peak at x = 0.5 and the next peak at x = 3.5, the period is 3.5 − 0.5 = 3. If instead you measured from a peak at 0.5 to the nearest trough, you’d get a value near 1.5 — exactly half, which you’d then double.

You can also estimate B backward from a graph: since T = 2π / |B|, rearranging gives |B| = 2π / T. A graph with period 3 therefore corresponds to |B| ≈ 2π/3 ≈ 2.094.

Key Takeaways

• The period of a sine function in the form y = A sin(Bx + C) + D is T = 2π / |B|.
• Only the coefficient B (the number multiplying x) affects the period; amplitude A, phase shift C, and vertical shift D do not.
• Use the absolute value of B, so negative coefficients give the same period as positive ones.
• Larger |B| shortens the period (more cycles); smaller or fractional |B| lengthens it (fewer cycles).
• The basic curve y = sin(x) has B = 1 and a period of 2π ≈ 6.283.
• On a graph, measure the horizontal distance between two consecutive matching peaks or troughs — peak-to-trough is only half a period.
• You can recover |B| from a graph using |B| = 2π / T.