The period of a repeating function is the horizontal length of one complete cycle: the smallest distance after which the graph repeats exactly. y = sinx and y = cosx have period 2π, or 360°, and y = tanx has period π. For y = sin(Bx) or y = cos(Bx), the period is 2π|B|.
In Spanish
período
Also called
period of a function, period of a trig function, fundamental period, cycle length
The period is the length of one full cycle. Multiplying x by 2 makes the cycles twice as short: 2π2 = π.
Formal definition and notation
A functionf is periodic if there is a positive numberP with f(x + P) = f(x) for every x in its domain. The smallest such P is the period. For y = Asin(Bx − C) + D and y = Acos(Bx − C) + D, the period is 2π|B|; the amplitude |A|, the phase shift and the vertical shift D do not change it. The number of cycles per unit, 1P, is the frequency.
Symbol
Read it as
Meaning
P
P
the period
f(x + P) = f(x)
f of x plus P equals f of x
shifting the graph P units gives the same graph
2π|B|
two pi over the absolute value of B
the period of sin(Bx) and cos(Bx)
Key facts
sinx, cosx, secx and cscx have period 2π; tanx and cotx have period π.
Larger |B| squeezes the graph: sin 4x has period π2.
|B| < 1 stretches it: sinx2 has period 4π.
The amplitude A changes the height, not the period.
Frequency is the reciprocal of period: a period of 0.5 seconds means 2 cycles per second.
Measure one period from any point to the next point where the graph starts the same pattern, such as peak to peak.
Period of sine and cosine
Period of sine and cosine
P = 2π|B| for y = Asin(Bx − C) + D
B
the number multiplying x
2π
the period of plain sinx and cosx (use 360° in degrees)
For tangent: P = π|B|. In degrees, replace 2π with 360° and π with 180°.
Period examples, step by step
Example 1: Period from B
Warm-up
Find the period of y = cos(4x).
Here B = 4.
P = 2π4 = π2.
Answer: The period is π2, which is 90°.
Example 2: Amplitude does not matter
Core
Find the period and amplitude of y = 3sin(x2).
Amplitude: |A| = 3, so the graph runs from −3 to 3.
B = 12, so P = 2π1/2 = 4π.
The 3 makes the wave taller; only B changes its length.
Answer: The amplitude is 3 and the period is 4π (that is, 720°).
Example 3: A Ferris wheel
Test-style
A rider’s height, in feet, on a Ferris wheel is h(t) = 20 − 18cos(π6t), where t is in minutes. How long does one full turn take?
Amplitude is half the vertical height of the wave, |A|. Period is the horizontal length of one cycle. For y = 3sin 2x: amplitude 3, period π.
Periodvs.Period in place value
In a large number, a period is a group of three digits separated by commas, such as the thousands period in 4,507,213. It has nothing to do with repeating graphs.
Periodvs.Repeating decimal period
The period of a repeating decimal is the repeating block of digits: 17 = 0.142857142857… has a period of 6 digits.
Quick check: 3 questions
Question 1 of 3
What is the period of y = sin(3x)?
Show answer and explanation
Answer: C.P = 2π3. 6π multiplies instead of dividing, 3 is B itself, and 2π ignores B.
Question 2 of 3
What is the period of y = tanx?
Show answer and explanation
Answer: A. Tangent repeats every π (180°). 2π is the period of sine and cosine, and π2 is where tangent has its first asymptote.
Question 3 of 3
What is the period of y = 5cos(2x) + 1?
Show answer and explanation
Answer: D. Only B = 2 matters: 2π2 = π. 5 is the amplitude, 1 is the vertical shift, and 10π wrongly multiplies by the amplitude.
Period: frequently asked questions
How do you find the period of a trig function?
Find the number B that multiplies x inside the function. For sine, cosine, secant and cosecant, divide 2π by |B|; for tangent and cotangent, divide π by |B|. For example, y = cos(5x − 1) + 7 has period 2π5. The other numbers shift or stretch the graph but do not change the period.
What is the difference between period and frequency?
The period is how long one cycle takes, and the frequency is how many cycles happen per unit, so each is the reciprocal of the other. A pendulum with a period of 2 seconds has a frequency of 12 cycle per second. A larger frequency always means a shorter period.
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