Period of a Sinusoid from k
Also known as period from k · k from period · 2 pi over k · period of a sine graph · horizontal stretch of a sine curve
Units aren’t used in this calculation — every value is a plain number.
Worked example: period 4 → k = π/2 = 1.5708 — press Try an example to run it live, then adjust anything.
Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value.
Learning zone
The period is the width of one full cycle, and it falls straight out of k: the sine function completes a revolution when its argument advances by 2π, so k(x − d) has to gain 2π, which takes 2π/k units of x. Worked example: y = 5 sin(πx/6) + 2 has k = π/6, so the period is 2π ÷ (π/6) = 12 — which is why that particular k turns up in every model of a monthly cycle over a year. Run it the other way for the more common task: a tide that repeats every 12.4 hours needs k = 2π/12.4 = 0.5067 radians per hour.
The relation is a reciprocal, so intuition about it has to be trained rather than assumed: doubling k halves the period, and a curve with k = 0.1 takes sixty-three units to complete one cycle. If your textbook works in degrees, the same reasoning gives period = 360/k instead, because a degree-measured sine completes its revolution at 360 rather than at 2π — the two conventions differ by the factor π/180 ≈ 0.01745, and mixing them is the single most common way a transformations answer comes out wrong by a factor of about 57. This page is written in radians, which is what the sine function in every calculator, spreadsheet and programming language actually consumes underneath. A negative k reflects the curve as well as setting the period; read the size of the answer and ignore its sign.
- = Period
- = Angular coefficient
- Period — Sinusoidal Model, Discriminant of a Quadratic
- Angular coefficient — Sinusoidal Model, Regression Slope from Correlation