5.3.4 Period and Frequency

A function that repeats over and over, as do sine and cosine, is called periodic, and the horizontal length of a pattern that repeats is called a period. The shortest repetition length is called the fundamental period. We can also think of the fundamental period as being the time required to complete a single cycle (if time is the independent variable). Thus, sine and cosine are periodic functions with fundamental period \(2\pi\). These functions are also periodic with period \(4\pi\), with period \(6\pi\), with period \(8\pi,...\). If there is no danger of confusion, "fundamental period" is often shortened to just "period." We adopt that convention in the rest of this section.

A concept closely related to period is frequency, which is the rate at which periods are being completed. Thus, if the unit of time is seconds, and a periodic function has period \(5\) seconds, then its frequency is \(1/5\) cycles per second. In general, frequency and period are reciprocals of each other. The unit for frequency is cycles per unit of time, and the unit for period is units of time per cycle.

Note 1 – Constant functions

Activity 4

For each of the following repeating phenomena, determine the period and frequency (in compatible time units) of whatever function describes the phenomenon.

1. Normal heart beat

2. Rotation of the Earth on its axis

3. Earth orbiting the Sun

4. Standard alternating current

5. Second hand on a clock

6. Minute hand on a clock

7. Hour hand on a clock

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