Period (Trig)
TrigonometryThe 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 itself.
Example
The basic $y = \sin(x)$ has a period of $2\pi \approx 6.28$ (in radians) or $360^\circ$. The pattern repeats every $6.28$ units along the $x$-axis.
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 it keeps repeating forever.
Definition
For $y = A\sin(Bx + C) + D$, the period $= 2\pi/|B|$. A larger $B$ compresses the graph horizontally (shorter period, faster oscillation). A smaller $B$ stretches it (longer period, slower oscillation). For tangent: period $= \pi/|B|$.
Example
$y = \sin(3x)$: period $= 2\pi/3 \approx 2.09$. This wave completes $3$ full cycles in the same space where $\sin(x)$ completes $1$. $y = \cos(x/2)$: period $= 2\pi/(1/2) = 4\pi \approx 12.57$. Much slower wave.
Key Insight
Period and frequency are reciprocals of each other. A short period means high frequency (many cycles per unit). A long period means low frequency (few cycles per unit). Music A4 ($440$ Hz) has a period of $1/440$ second.
Definition
The period $T$ of a periodic function satisfies $f(x + T) = f(x)$ for all $x$. The fundamental period is the smallest such $T > 0$. For $\sin(Bx)$, $T = 2\pi/|B|$; for $\tan(Bx)$, $T = \pi/|B|$. In Fourier analysis, a function with fundamental period $T$ has Fourier series with frequencies $n/T$ for non-negative integers $n$, corresponding to harmonics of the fundamental frequency $1/T$.
Example
$\sin^2(x)$ has fundamental period $\pi$, not $2\pi$, because $\sin^2(x) = (1 - \cos(2x))/2$, which has period $\pi$. This reduction in period arises because squaring eliminates the sign distinction between positive and negative half-cycles.
Key Insight
The relationship between period and the coefficient $B$ is a scaling law: replacing $x$ by $Bx$ compresses the domain by factor $B$, which maps the standard period $2\pi$ to $2\pi/B$. This is the same scaling that transforms time to frequency in the Fourier transform, making period/frequency duality fundamental to all of harmonic analysis.