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Chapter 4: Problem 94

What is a periodic function? Why are the sine and cosine functions periodic?

Short Answer

Expert verified
A periodic function is a function that repeats its values at regular intervals, known as periods. The sine and cosine functions are examples of periodic functions because they repeat their values every \(2\pi\) radians or 360 degrees, which is their period.

Step by step solution

01

Definition of Periodic Function

A periodic function is a function that repeats its values in regular intervals or 'periods'. Mathematically, function \(f(x)\) is said to be periodic if there exists a non-zero number \(P\) such that \(f(x+P) = f(x)\) for every \(x\) in the domain of \(f(x)\). \(P\) here is the period of the function. It is the smallest positive number for which the function is periodic.
02

Understand the Sine Function

The sine function, represented as \(sin(x)\), is a periodic function because it repeats its values every \(2\pi\) radians, or 360 degrees. If we observe a sine wave, it goes through one cycle of its pattern from 0 to \(2\pi\), and this pattern repeats for every increase of \(2\pi\). So, mathematically, we can say, \(sin(x + 2\pi) = sin(x)\) for all \(x\). Hence, the period of \(sin(x)\) is \(2\pi\).
03

Understand the Cosine Function

Similar to the sine function, the cosine function, \(cos(x)\), is also a periodic function and it repeats its values every \(2\pi\) radians, or 360 degrees. This happens because a cosine wave also completes one full cycle of its pattern from 0 to \(2\pi\), and this pattern repeats for every \(2\pi\) increase. Hence, we have \(cos(x + 2\pi) = cos(x)\) for all \(x\). So, just like the sine function, the period of the cosine function is also \(2\pi\).

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After using the four-step procedure to graph \(y=-\cot \left(x+\frac{\pi}{4}\right),\) I checked my graph by verifying it was the graph of \(y=\cot x\) shifted left \(\frac{\pi}{4}\) unit and reflected about the \(x\) -axis.

Find the slant asymptote of $$ f(x)=\frac{2 x^{2}-7 x-1}{x-2} $$ (Section \(2.6, \text { Example } 8)\)

Use the keys on your calculator or graphing utility for converting an angle in degrees, minutes, and seconds \(\left(D^{\circ} M^{\prime} S^{\prime \prime}\right)\) into decimal form, and vice versa. Convert each angle to a decimal in degrees. Round your answer to two decimal places. $$65^{\circ} 45^{\prime} 20^{\prime \prime}$$

What does a phase shift indicate about the graph of a sine function? How do you determine the phase shift from the function's equation?

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=-4 \cos \left(2 x-\frac{\pi}{2}\right)$$
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