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Period Calculator

Compute period, amplitude, and frequency of trigonometric functions

Select Function Type

Sine a*sin(kx) Cosine a*cos(kx) Tangent a*tan(kx)

f(x)=a*sin(kx)

f(x)=sin(x)

Period and Amplitude

Sine/Cosine: period T = 2pi/|k|, amp = |a|

Tangent: period T = pi/|k|, undefined amp

Frequency f = 1/T (periods per unit)

Angular frequency omega = 2pi/T

The period of a function is the length of one complete cycle. Trigonometric functions repeat infinitely. The coefficient k inside the function affects the horizontal stretch (period). The coefficient a outside affects vertical stretch (amplitude for sine/cosine).

⚠ Tangent has no amplitude (goes to infinity). Its range is all real numbers. Only sine and cosine have bounded amplitude.

What Are Period and Amplitude?

Periodic functions repeat at regular intervals. The period is the smallest T with f(x+T)=f(x). The amplitude is half the distance between maximum and minimum (for bounded functions). Frequency = 1/period measures how often the function repeats per unit.

Sine/Cosine

T=2pi/|k|, amp=|a|. Period shortens as k increases. f(x)=sin(x): T=2pi. sin(2x): T=pi.

Tangent

T=pi/|k| (shorter than sine). No amplitude (unbounded). Vertical asymptotes at x=(2n+1)pi/(2k).

Frequency

f=1/T. Angular freq omega=2pi/T=k. Higher k means more cycles per unit length.

Phase Shift

f(x)=a*sin(kx+phi) has horizontal shift -phi/k. Does NOT affect period or amplitude.

Teaching Example: f(x)=2*sin(3x). Amplitude=|2|=2. Period=2pi/3. Frequency=3/(2pi). Range=[-2,2]. The graph completes 3 cycles in the length of one standard sine period.

Applications

Trigonometry Physics Signal Processing Engineering Sound Waves

Frequently Asked Questions

What is period?▼

Smallest T with f(x+T)=f(x). Sine: 2pi/k. Tangent: pi/k. Lower k = longer period.

Find period of sin(kx)?▼

T=2pi/|k|. Example: sin(x) has 2pi, sin(2x) has pi. Higher k compresses the graph horizontally.

What is amplitude?▼

|a| for sine/cosine. Half the vertical distance between max and min. Tangent has no amplitude.

Period and frequency?▼

f=1/T. Angular freq = 2pi/T = k. Higher k = higher frequency = more oscillations per unit.

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