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Recipe

Find the fundamental period of a sum of periodic terms

Use when: you have a sum of two or more periodic terms, each with its own known period, and need the period of the whole combination

Goal: the smallest positive number that is a common period of every term, found without graphing

Taught in Periodic functions and even/odd symmetry, revisited

Quick recipe

  1. Find each term's own fundamental period
  2. Write every period as a multiple of one common unit
  3. Find the smallest common multiple of the individual periods
  4. Confirm no smaller positive number works for every term
  1. 1

    For a term like \cos(bx) or \sin(bx), the period is \frac{2\pi}{|b|}; for any other periodic piece, use its own known period directly.

    Example

    \sin(4x) has b=4, so its period is \frac{2\pi}{4}=\frac{\pi}{2}. \cos(6x) has b=6, so its period is \frac{2\pi}{6}=\frac{\pi}{3}.

    Why: the sum's own period can only be built from the periods already present in each individual term.

  2. 2

    Express each period as a number of units of \pi, so the periods become plain numbers ready to compare.

    Example

    \frac{\pi}{2} = \frac12\cdot\pi and \frac{\pi}{3}=\frac13\cdot\pi, so the numbers to compare, in units of \pi, are \frac12 and \frac13.

    Why: comparing how many times one period fits into a candidate only makes sense once every period is measured in the same unit.

  3. 3

    Find the smallest positive number that is a whole-number multiple of every individual period at once.

    Example

    \frac12's multiples are \frac12,1,\frac32,2,\dots and \frac13's multiples are \frac13,\frac23,1,\frac43,\dots; the smallest number on both lists is 1, so the period is 1\cdot\pi=\pi.

    • every period is a rational multiple of the common unit — a smallest common multiple exists, and it's the fundamental period.
    • the ratio between two periods is irrational — no common period exists, and the sum isn't periodic at all.

    Why: a shared period has to complete a whole number of cycles of every single term, or some term would be left mid-cycle.

  4. 4

    Check that nothing smaller than the candidate from Step 3 is also a whole-number multiple of every individual period.

    Example

    No number smaller than 1 (in units of \pi) is a whole multiple of both \frac12 and \frac13, so \pi is genuinely the smallest, not just a common one.

    Why: a common period always has infinitely many multiples that also work, but the fundamental period is specifically the smallest positive one, by definition.

Common mistakes

  • Adding the two periods together instead of finding their smallest common multiple.
  • Assuming the sum's period always equals the smaller of the individual periods.
  • Finding a common multiple but skipping the check for a smaller one that also works.
  • Assuming every sum of periodic functions is itself periodic, when an irrational ratio of periods rules that out.