Period (Trig)
The period of a trigonometric function is the length of one complete cycle of the graph.
Formula
\text{period} = \frac{2\pi}{|B|} \text{ for } y = A\sin(Bx + C) + D
Definition
The period of a sine or cosine graph is how long it takes for the wave to complete one full up-down-up pattern and repeat. Formally, a periodic function satisfies $f(x + T) = f(x)$ for all $x$, and the fundamental period is the smallest such $T > 0$; for $y = A\sin(Bx + C) + D$, the period $= 2\pi/|B|$ (or $\pi/|B|$ for tangent), so a larger $B$ compresses the graph horizontally into a shorter, faster period, while a smaller $B$ stretches it. In Fourier analysis, a function with fundamental period $T$ has a Fourier series with frequencies $n/T$, harmonics of the fundamental frequency $1/T$.
Example
The basic $y = \sin(x)$ has period $2\pi \approx 6.28$ (or $360^\circ$), while $y = \sin(3x)$ has period $2\pi/3 \approx 2.09$, completing $3$ cycles in the space $\sin(x)$ takes for $1$. Interestingly, $\sin^2(x)$ has fundamental period $\pi$, not $2\pi$, because $\sin^2(x) = (1 - \cos(2x))/2$, which has period $\pi$, a reduction that arises because squaring eliminates the sign distinction between positive and negative half-cycles.
Key Insight
Think of a period like a song that loops: after one period, the wave looks exactly the same as where it started, and repeats forever; period and frequency are reciprocals, so a short period means high frequency and a long period means low frequency (music A4 at $440$ Hz has a period of $1/440$ second). Replacing $x$ by $Bx$ compresses the domain by factor $B$, mapping the standard period $2\pi$ to $2\pi/B$, the same scaling that transforms time to frequency in the Fourier transform, making period/frequency duality fundamental to harmonic analysis.