Periodic Functions

Find the fundamental period of standard, transformed, composite and combined functions, and know when the LCM rule fails.

Builds on: Part 17 · Even and Odd Functions, Part 16 · Greatest Integer, Fractional Part, Signum and Max-Min.

Video coming soon

Standard periods

Learn these

sin, cos, sec, cosec: 2π; tan, cot: π; |sin x|, sin²x: π; {x}: 1; f(ax + b): T/|a|.

Table of periods. — Even powers and moduli halve 2π.

sin x and sin 3x. — Shaded windows of length 2π/3.

Transformed

sin 3x vs sin x

sin(3x + 2) has period 2π/3: three waves where sin x has one. The shift doesn't change the period.

Exception

|sin x| + |cos x|

The LCM of the two periods is π, but f(x + π/2) = |cos x| + |sin x| = f(x): the fundamental period is π/2.

Summary

Key formulas

period of f (a x + b) = \frac{T}{∣ a ∣}

Period of transformed functions

Worked examples

From the video

1. Period of sin 2x + cos 3x.

2π.

2. Period of |sin x| + |cos x|.

π/2.

3. Period of sin⁴x + cos⁴x.

π/2.

4. Period of cos(sin x).

π.

JEE-style question

Your turn

The fundamental period of |sin 2x| + |cos 2x| is:

(a)π/4

(b)π/2

(c)π

(d)2π

Show the answer and the traps

|sin x| + |cos x| has period π/2. Replacing x by 2x divides it by 2: π/4. Option (a).

(b) is the period of each part, |sin 2x|, without testing a smaller value. (c) ignores the modulus: it's the period of sin 2x.

(d) is the period of sin x itself, ignoring both the modulus and the 2x.

Watch out

Common mistakes

Taking the LCM as the final answerTest smaller candidates when there are moduli or even powers.

Letting the shift b change the period of f(ax + b)Only a matters: T/|a|.

Assuming every sum of periodic functions is periodicsin x + {x} is not periodic.

Practice

Try these

1. Period of sin(4x/3).

3π/2.

2. Period of sin 3x + cos 2x.

2π.

3. Period of tan x + cot x.

π.

4. Period of {3x} + sin(πx).