Frequency
Frequency is the number of complete cycles of a periodic function per unit interval, equal to the reciprocal of the period.
Formula
\text{frequency} = \frac{1}{\text{period}} = \frac{|B|}{2\pi}
Definition
Frequency tells you how many full wave cycles happen per unit of time or distance; high frequency means the wave repeats quickly, low frequency means it repeats slowly. For $y = A\sin(Bx + C) + D$, frequency $= |B|/(2\pi) = 1/\text{Period}$; in physics it is measured in hertz (Hz), cycles per second, with $y = A\sin(2\pi ft)$ for frequency $f$. Angular frequency $\omega = 2\pi f = 2\pi/T$ determines how rapidly a sinusoid oscillates, and the Fourier transform $F(\omega) = \int f(t)e^{-i\omega t}\, dt$ decomposes a signal into its frequency components.
Example
A wave with period $2$ seconds has frequency $= 0.5$ cycles per second, while one with period $0.01$ seconds has frequency $100$ Hz. Middle C on a piano is $261.63$ Hz, so its wave $y = A\sin(2\pi \cdot 261.63t)$ completes $261.63$ cycles every second. Audio CDs sample at $44{,}100$ Hz, supporting frequencies up to $22{,}050$ Hz, chosen to satisfy the Nyquist-Shannon sampling theorem across the entire audible range (up to about $20{,}000$ Hz).
Key Insight
Higher pitch sounds have higher frequency: a dog whistle and a foghorn are both sine waves, just at very different frequencies, and the factor $B$ in $\sin(Bx)$ is the angular frequency, measured in radians per unit, which is why $2\pi$ appears in the period formula. In quantum mechanics, frequency is proportional to energy via $E = hf$ (Planck's relation), making frequency a bridge between the wave description (classical) and particle description (quantum) of light and matter.