Graph the following functions. Which are periodic?...

Graph the following functions. Which are periodic? quasi-periodic? If periodic, what is the (minimal) period? (a) $\sin 4 t+\cos 6 t$, (b) $1+\sin \pi t$, (c) $\cos \frac{1}{2} \pi t+\cos \frac{1}{3} \pi t$, (d) $\cos t+\cos \pi t$, (e) $\sin \frac{1}{4} t+\sin \frac{1}{5} t+\sin \frac{1}{6} t$, (f) $\cos t+\cos \sqrt{2} t+\cos 2 t$, (g) $\sin t \sin 3 t$.

Key Concepts

Periodic Function

A periodic function is one that repeats its values at regular intervals, meaning there exists a positive constant T (the period) such that for all t, f(t + T) = f(t). This concept is central in understanding functions like sine and cosine, and in analyzing sums of such functions to determine their overall repeating behavior.

Minimal Period

The minimal period of a periodic function is the smallest positive number T for which the function repeats. When dealing with the sum of periodic functions, if the individual periods are commensurate (i.e., their ratio is rational), the minimal period of the sum is typically the least common multiple (LCM) of the individual periods.

Quasi-Periodic Function

A quasi?periodic function results when a system is composed of two or more periodic functions whose periods are incommensurate (their ratio is irrational). In such cases, although the component functions individually repeat, the overall function does not have a unique minimal period and displays complex behavior that never exactly repeats itself.

Sums of Trigonometric Functions

When adding trigonometric functions like sine and cosine, each with its own angular frequency (and hence period), the overall periodicity depends on the relationship between these frequencies. If the periods have a rational ratio, a common period exists and the sum is periodic; otherwise, the sum may be quasi?periodic.

Frequency and Angular Frequency

Angular frequency is the coefficient multiplying the variable in the argument of trigonometric functions, directly influencing the oscillation rate of the function. The period is calculated as the reciprocal of the angular frequency (appropriately scaled), and differences in angular frequencies among summed functions play a key role in determining the overall periodicity.

Rational Ratio Condition

For the sum of two or more periodic functions to be periodic, it is necessary that the ratio of their individual periods (or angular frequencies) be rational. This condition ensures that the functions have a common repeating interval, allowing the overall function to have a well-defined minimal period.

Transcript

00:01 The first function is 4 sign 5 t plus 3 sign 60.

00:12 We need to find the period of this function, fundamental period.

00:16 Remember, it's a nice formula.

00:20 If fundamental period, fp is fundamental period of a function f of t is capital t, and the fundamental period of f of kt, where k sums scale up is t by mod k, k not equal to now we know that the fundamental period of sine t is 2 pi.

00:47 So what will be fundamental period of sine pi t? it is 2 pi by 5.

00:55 4 is a constant so it doesn't matter.

00:57 Similarly what is the fundamental period of sine 60? a fundamental period of sign 60 is 2 pi by 6.

01:06 Now since both the functions are added we need to take the lcm of both the periods 2 pi by 5 and 2 pi by 6 2 pi is a constant into lcm of 1 by 5 comma 1 by 6 the lcm of fractions is defined as lcm of numerators that is 1 comma 1 and hcf of denominators that is 5 6 what is lcm of 1 comma 1 it is 1 and hcf of 5 comma 6 is also 1 so answer is 2 pi so the fundamental period of the first function is 2 pi the next signal is 12 plus 12 plus 5 cost 2 t plus 2 cost 40 plus 45 degrees.

01:55 5 cost 2 t plus 4 t plus 45 degrees.

02:09 As usual, 12 is a constant so you can ignore.

02:12 What is the period of cost 2 t is 2 pi by 2? and what is the period of cost 4 t plus 45? remember horizontal shift horizontal translation will not change the period of the function.

02:28 So plus 45 degrees doesn't matter at all...

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