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Chapter 6: Problem 65

Show that the given function is periodic with period less than \(2 \pi\). [Hint: Find a positive number \(k\) with \(k<2 \pi \text { such that } f(t+k)=f(t) \text { for every t in the domain of } f .]\) $$f(t)=\sin (\pi t)$$

Short Answer

Expert verified
Question: Show that the given function \(f(t) = \sin(\pi t)\) is periodic with a period less than \(2\pi\). Answer: The given function, \(f(t) = \sin(\pi t)\), is periodic with period \(k=2\), which is less than \(2\pi\).

Step by step solution

01

Taking help of hint to find k

Since we need to find a positive number \(k\), such that \(k < 2\pi\), let's make use of the hint provided: \(f(t+k) = f(t)\), i.e. \(\sin(\pi(t+k)) = \sin(\pi t)\). To deduce the value of \(k\), let's set \(\pi(t+k)+2\pi\m = \pi t\) for some integer \(\m\).
02

Setting the expression for k

Let's substitute \(\pi(t+k)+2\pi\m = \pi t\) in the equation we have postulated above. Now, we get $$\pi(t+k)+2\pi\m = \pi t.$$.
03

Simplifying the equation

Divide every term by \(\pi\) to simplify further: $$t+k+2m = t.$$
04

Finding the value of k

Now, we can solve for \(k\): $$k+2m = 0.$$ To get the smallest positive value of \(k<2\pi\), set \(m=-1\). Then \(k=2\).
05

Checking if \(f(t+k)=f(t)\) with our found k

To ensure that our found value of \(k=2\) satisfies the condition \(f(t+k)=f(t)\), let's substitute it in the original function with both \(t\) and \(t+k\): $$f(t) = \sin(\pi t),$$ $$f(t+k) = \sin(\pi (t+2)).$$
06

Using periodic property of sine function

Since the sine function has a period of \(2\pi\), we can say that \(\sin(\theta+2\pi)=\sin(\theta)\) for any \(\theta\). Thus we have, $$\sin(\pi (t+2)) = \sin(\pi t+2\pi).$$
07

Concluding the periodicity of given function

As \(\sin(\theta+2\pi)=\sin(\theta)\) for any \(\theta\), we can say that: $$\sin(\pi t+2\pi) = \sin(\pi t),$$ which implies that $$f(t+k) = f(t).$$ Thus, we can conclude that the given function, \(f(t) = \sin(\pi t)\), is periodic with period \(k=2\), which is less than \(2\pi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sine Function
The sine function is one of the most fundamental trigonometric functions. It expresses how the angle in a right-angled triangle relates to the ratio of the length of the opposite side to the hypotenuse. It can be defined as follows:
  • For an acute angle in a right-angled triangle, the sine is the ratio of the length of the side opposite the angle to the hypotenuse.
  • In the unit circle, it represents the y-coordinate of a point corresponding to an angle on the circle.
Mathematically, the sine function is expressed as \( \sin(\theta) \). It's a periodic wave, which means it repeats its values at regular intervals.
The sine function takes any real number as input, and outputs a value between -1 and 1, making it a bounded function. This feature is graphically represented as a smooth wave, starting from the origin, oscillating between its maximum and minimum values.
Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It is essential for the understanding of periodic functions like sine and cosine. Let's break it down a bit:
  • Angles and Triangles: It primarily focuses on right-angled triangles, but its principles apply to any triangle.
  • Functions: The primary trigonometric functions are sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides.
Trigonometry is incredibly useful in various fields such as physics, engineering, and even in graphic design. It's used for anything involving cyclical patterns, waves, or rotations. Without these concepts, our understanding of periodic phenomena, like sound waves and light, would be incomplete.
Trigonometry also extends beyond simple triangle measurements and helps in understanding the properties of waves, oscillations, and many real-world functions that repeat themselves, such as daily temperature variations and alternating current in electricity.
Period of a Function
The period of a function is the interval at which the function repeats its values. For trigonometric functions, such as the sine function, this is a crucial aspect because it describes how the function behaves across its domain.
  • Sine Function: The sine function has a standard period of \(2\pi\), meaning it repeats every \(2\pi\) radians.
  • Finding the Period: To find a function's period, you look for the smallest positive value, \(k\), such that \(f(t+k) = f(t)\) for all possible \(t\).
In the exercise, the period is calculated to be 2, which means that the function \(f(t) = \sin(\pi t)\) repeats its values every 2 units. This is less than the typical \(2\pi\) for the basic sine function, demonstrating how periods can vary with transformations.
Understanding the period of a function helps in predicting its future behavior and is essential in applications involving cycles, such as sound waves, seasonal trends, and electrical circuits. By knowing when a function repeats, you can simplify analyses and calculations by focusing on just one complete cycle.

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Most popular questions from this chapter

Graph the function. Does the function appear to be periodic? If so, what is the period? $$g(t)=|\sin t|$$

Provide further examples of functions with different graphs, whose graphs appear identical in certain viewing windows. Approximating trigonometric functions by polynomials. For each odd positive integer \(n,\) let \(f_{n}\) be the function whose rule is $$ f_{n}(t)=t-\frac{t^{3}}{3 !}+\frac{t^{5}}{5 !}-\frac{t^{7}}{7 !}+\cdots-\frac{t^{n}}{n !} $$ since the signs alternate, the sign of the last term might be \+ instead of \(-,\) depending on what \(n\) is. Recall that \(n !\) is the product of all integers from 1 to \(n\); for instance, \(5 !=1 \cdot 2 \cdot 3 \cdot 4 \cdot 5=120\) (a) Graph \(f_{7}(t)\) and \(g(t)=\sin t\) on the same screen in a viewing window with \(-2 \pi \leq t \leq 2 \pi .\) For what values of \(t\) does \(f_{7}\) appear to be a good approximation of \(g ?\) (b) What is the smallest value of \(n\) for which the graphs of \(f_{n}\) and \(g\) appear to coincide in this window? In this case, determine how accurate the approximation is by finding \(f_{n}(2)\) and \(g(2)\)

The table below shows the number of unemployed people in the labor force (in millions) for \(1984-2005 .\) (a) Sketch a scatter plot of the data, with \(x=0\) corresponding to 1980 (b) Does the data appear to be periodic? If so, find an appropriate model. (c) Do you think this model is likely to be accurate much beyond \(2005 ?\) Why? \(\begin{array}{|c|c|}\hline \text { Year } & \text { Unemployed } \\\\\hline 1984 & 8.539 \\\\\hline 1985 & 8.312 \\\\\hline 1986 & 8.237 \\\\\hline 1987 & 7.425 \\\\\hline 1988 & 6.701 \\\\\hline 1989 & 6.528 \\\\\hline 1990 & 7.047 \\\\\hline 1991 & 8.628 \\\\\hline 1992 & 9.613 \\\\\hline 1993 & 8.940 \\\\\hline 1994 & 7.996 \\\\\hline\end{array}\) \(\begin{array}{|c|c|}\hline \text { Year } & \text { Unemployed } \\\\\hline 1995 & 7.404 \\\\\hline 1996 & 7.236 \\\\\hline 1997 & 6.739 \\\\\hline 1998 & 6.210 \\\\\hline 1999 & 5.880 \\\\\hline 2000 & 5.692 \\\\\hline 2001 & 6.801 \\\\\hline 2002 & 8.378 \\\\\hline 2003 & 8.774 \\\\\hline 2004 & 8.149 \\\\\hline 2005 & 7.591 \\\\\hline\end{array}\)

In Exercises \(49-54\), prove the given identity. $$\tan t=\frac{1}{\cot t}[\text {Hint}: \text { See page } 497]$$

In Exercises \(55-60\), find the values of all six trigonometric functions at \(t\) if the given conditions are true. $$\begin{aligned} &\cos t=-1 / 2 \quad \text { and } \quad \sin t>0\\\ &\text { [Hint: }\left.\sin ^{2} t+\cos ^{2} t=1 .\right] \end{aligned}$$
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