Period (of a Trig Function)

The period of a function is the smallest positive $T$ such that $f(x + T) = f(x)$ for all $x$. The function repeats itself every $T$ units along the input axis.

Standard trig periods:

• $\sin x$, $\cos x$: period $2\pi$ (one full circle).
• $\tan x$, $\cot x$: period $\pi$ (half a circle — tan repeats faster because of how it's defined as ratio).
• $\csc x$, $\sec x$: period $2\pi$.

For a transformed sinusoid $A\sin(Bx + C) + D$:

• Amplitude = $|A|$ (peak height).
• Period = $\frac{2\pi}{|B|}$ (a larger $|B|$ compresses the wave).
• Phase shift = $-C/B$ (horizontal shift).
• Vertical shift = $D$.

Period is the central concept in frequency analysis, sound waves (Hz = cycles per second = $1/T$), planetary orbits, AC current, and Fourier series — which decompose any periodic function into a sum of sines and cosines of varying period.