(a) The fundamental period is 2 π/ 2 π=1. (b) The fundamental period is 2 π/(4 / L)=(1)/(2) πL.

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(a) The fundamental period is 2 π/ 2 π=1. (b) The...

(a) The fundamental period is $2 \pi / 2 \pi=1.$ (b) The fundamental period is $2 \pi /(4 / L)=\frac{1}{2} \pi L.$ (c) The fundamental period of $\sin x+\sin 2 x$ is $2 \pi.$ (d) The fundamental period of $\sin 2 x+\cos 4 x$ is $2 \pi / 2=\pi.$ (e) The fundamental period of $\sin 3 x+\cos 4 x$ is $2 \pi$ since the smallest integer multiples of $2 \pi / 3$ and $2 \pi / 4=\pi / 2$ that are equal are 3 and $4,$ respectively. (f) The fundamental period of $f(x)$ is $2 \pi /(n \pi / p)=2 p / n$.

(a) The fundamental period is $2 \pi / 2 \pi=1.$
(b) The fundamental period is $2 \pi /(4 / L)=\frac{1}{2} \pi L.$
(c) The fundamental period of $\sin x+\sin 2 x$ is $2 \pi.$
(d) The fundamental period of $\sin 2 x+\cos 4 x$ is $2 \pi / 2=\pi.$
(e) The fundamental period of $\sin 3 x+\cos 4 x$ is $2 \pi$ since the smallest integer multiples of $2 \pi / 3$ and $2 \pi / 4=\pi / 2$ that are equal are 3 and $4,$ respectively.
(f) The fundamental period of $f(x)$ is $2 \pi /(n \pi / p)=2 p / n$.

Key Concepts

Least Common Multiple for Periods

When dealing with sums of periodic functions, each function may have its own period. The overall function is periodic if there is a common period, which is typically determined by finding the least common multiple (LCM) of the individual periods. This LCM is the smallest positive value for which all component functions repeat their cycles, thus ensuring the sum is periodic.

Composite Function Periodicity

Composite functions, particularly those involving trigonometric sums or combinations, require careful analysis to determine their overall periodicity. Techniques include identifying individual periods, adjusting for phase shifts or amplitude changes, and then computing the LCM of these periods to find the fundamental period of the overall function.

Trigonometric Function Periodicity

Trigonometric functions such as sine and cosine are inherently periodic, with standard periods that can be modified by changes in their arguments. For example, a sine function of the form sin(kx) has a fundamental period of 2?/|k|. Understanding this helps in determining the behavior of more complex functions involving trigonometric components.

Fundamental Period

The fundamental period is defined as the smallest positive value P for which a periodic function f(x) satisfies f(x + P) = f(x) for all x. It represents the interval over which the function’s behavior repeats, and finding it is essential in understanding the structure of periodic functions.

Frequency and Period Relationship

In the context of trigonometric functions, frequency refers to the number of cycles the function completes in a unit interval, and is inversely related to the period. Specifically, if a function has an angular frequency k, then its period is given by 2? divided by the absolute value of k, highlighting the inverse relationship between frequency and period.

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