Frequency
TrigonometryFrequency 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 per unit along the $x$-axis). High frequency means the wave repeats quickly; low frequency means it repeats slowly.
Example
A wave with period $2$ seconds has frequency $= 1/2 = 0.5$ cycles per second. A wave with period $0.01$ seconds has frequency $= 100$ cycles per second ($100$ Hz).
Key Insight
Higher pitch sounds have higher frequency. A dog whistle (very high frequency) and a foghorn (very low frequency) are both sine waves, just with very different frequencies.
Definition
For $y = A\sin(Bx + C) + D$, the frequency is $|B|/(2\pi)$, and the period is $2\pi/|B|$. Frequency $= 1/\text{Period}$. In physics, frequency is measured in hertz (Hz) $=$ cycles per second, and the sine function is written $y = A\sin(2\pi ft)$ where $f$ is frequency in Hz.
Example
Middle C on a piano is $261.63$ Hz. Its wave: $y = A\sin(2\pi \cdot 261.63t)$. Period $= 1/261.63 \approx 0.00382$ seconds. In $1$ second, $261.63$ complete cycles occur.
Key Insight
The factor $B$ in $\sin(Bx)$ is the angular frequency ($\omega$), measured in radians per unit. It relates to regular frequency $f$ by $\omega = 2\pi f$. This is why $2\pi$ appears in the period formula: one cycle $= 2\pi$ radians of the sine function.
Definition
Angular frequency $\omega = 2\pi f = 2\pi/T$ determines how rapidly a sinusoidal function oscillates. The Fourier transform $F(\omega) = \int f(t)e^{-i\omega t}\, dt$ decomposes a signal into its frequency components. The Nyquist-Shannon sampling theorem states a signal with maximum frequency $f_{\max}$ can be perfectly reconstructed from samples at rate $> 2f_{\max}$.
Example
Audio CDs sample at $44{,}100$ Hz, supporting frequencies up to $22{,}050$ Hz (slightly above the human hearing limit of $\sim 20{,}000$ Hz). This sampling rate is chosen to satisfy the Nyquist criterion for the entire audible frequency range.
Key Insight
The Fourier transform is the continuous version of the discrete Fourier transform. 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.