How to find the period of a sine or cosine function

Key concept: If a trig graph is written in the form $y=A\sin(Bx-C)+D$ or $y=A\cos(Bx-C)+D$, the period comes from the coefficient of $x$ inside the function. That small detail controls how fast the wave repeats.

Step by step

For a sine or cosine function in the form

$y=A\sin(Bx-C)+D \quad \text{or} \quad y=A\cos(Bx-C)+D,$

the fundamental period is

$\frac{2\pi}{|B|}, \quad B\ne 0.$

Why this works

The basic graphs $\sin x$ and $\cos x$ repeat every $2\pi$. When the input changes to $Bx$, the graph is horizontally compressed or stretched by a factor of $\frac{1}{|B|}$. So the repeat length changes from $2\pi$ to

$\frac{2\pi}{|B|}.$

Quick check

• If $B=1$, period is $2\pi$
• If $B=2$, period is $\pi$
• If $B=\frac{1}{2}$, period is $4\pi$

Important note

The sign of $B$ does not change the period. A negative $B$ only reflects the graph horizontally; the period is still based on $|B|$.

Pitfall alert

A common mistake is to confuse the period formula with the phase shift. The value of $C$ changes where the graph starts, but it does not change how long one full cycle takes. Another slip is forgetting the absolute value: $B=-3$ still gives period $\frac{2\pi}{3}$, not a negative period.

Try different conditions

If the function is written with degrees instead of radians, the pattern changes to $\frac{360^\circ}{|B|}$. For example, $y=\sin(4x)$ in degree mode has period $90^\circ$, while in radian mode $y=\sin(4x)$ has period $\frac{\pi}{2}$. The same idea applies: the coefficient inside the trig function controls repetition speed.

Further reading

fundamental period, angular frequency, horizontal stretch