Definition: The period of a repeating function is the length of the shortest interval after which its graph repeats; for sine and cosine with y = sin(Bx) it is 2 pi over |B|.
What is the period?
A periodic function repeats its pattern. The period is the horizontal length of one full cycle. Basic sine and cosine have period 2 pi. For y = A sin(Bx) or y = A cos(Bx), the period is 2 pi / |B|. For tangent it is pi / |B|.
How do you use it?
Worked example: y = 5 cos(2x3). Here B = 2/3, so the period is 2 pi divided by 2/3, which equals 2 pi times 3/2 = 3 pi.
Going the other way, to get a period of 8 minutes in y = sin(Bt), solve 2 pi / B = 8, so B = pi/4.
What do students mistake?
Multiplying instead of dividing, which would give 2 pi times B. A bigger B squeezes the wave and makes the period shorter, so B belongs in the denominator. Mathbot pictures a faster-spinning wheel.
Where to learn it
- Graphing Waves: Sine and Cosine Models – lesson
- Graphing Sine, Cosine and Tangent Functions – lesson