Chapter 10: Problem 1
In each of Problems 1 through 8 determine whether the given function is periodic. If so, find its fundamental period. $$ \sin 5 x $$
Short Answer
Expert verified
Answer: Yes, the function sin(5x) is periodic. Its fundamental period is (2π)/5.
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fundamental Period
In the world of periodic functions, the concept of the fundamental period is vital. It represents the shortest interval in which the function begins to repeat itself. To put it simply, for any periodic function, the fundamental period is the length of one complete cycle.
For instance, imagine taking a walk along a path that winds in a perfect loop. No matter how far you walk, every section of the path you traverse repeats after you complete one loop. This loop is like the fundamental period of the path. Similarly, with functions like \(\sin 5x\), we are looking for that 'loop' or the shortest distance along the x-axis after which the function values start repeating. In the exercise provided, finding the fundamental period helps us understand this repeating cycle for the function \(\sin 5x\).
For instance, imagine taking a walk along a path that winds in a perfect loop. No matter how far you walk, every section of the path you traverse repeats after you complete one loop. This loop is like the fundamental period of the path. Similarly, with functions like \(\sin 5x\), we are looking for that 'loop' or the shortest distance along the x-axis after which the function values start repeating. In the exercise provided, finding the fundamental period helps us understand this repeating cycle for the function \(\sin 5x\).
Trigonometric Functions
Trigonometric functions are the bread and butter of periodicity. These functions, which include the sine, cosine, and tangent functions, originate from the relationships of angles and sides of a right triangle. Moreover, they have applications extending far beyond geometry, into various fields like physics, engineering, and even music.
One of the unique properties of trigonometric functions is their periodic nature. This means they repeat their values in a predictable pattern. When graphed, they produce waves that continue indefinitely, in both the positive and negative directions along the x-axis. Because these functions are periodic, they have a fundamental period. For the basic \(\sin(x)\) function, this period is \(2\pi\), signifying one complete oscillation from 0 to \(2\pi\) before the pattern repeats.
One of the unique properties of trigonometric functions is their periodic nature. This means they repeat their values in a predictable pattern. When graphed, they produce waves that continue indefinitely, in both the positive and negative directions along the x-axis. Because these functions are periodic, they have a fundamental period. For the basic \(\sin(x)\) function, this period is \(2\pi\), signifying one complete oscillation from 0 to \(2\pi\) before the pattern repeats.
Sine Function
Diving into the specifics, the sine function is particularly special among its trigonometric relatives. When plotted, it forms a smooth and continuous wave known for its 'sine curve' shape. The standard sine function, \(\sin(x)\), has a fundamental period of \(2\pi\), which corresponds to a full rotation around the unit circle.
But the story doesn't end there. When we introduce a multiplier to the variable, such as in \(\sin(5x)\), we effectively alter the frequency of the wave. In this modification, the function will complete its cycle more rapidly. It's like listening to a song on fast forward; the notes (or in our case, the values of the function) come more quickly, but the melody (periodicity) remains the same.
But the story doesn't end there. When we introduce a multiplier to the variable, such as in \(\sin(5x)\), we effectively alter the frequency of the wave. In this modification, the function will complete its cycle more rapidly. It's like listening to a song on fast forward; the notes (or in our case, the values of the function) come more quickly, but the melody (periodicity) remains the same.
Analyzing Periodicity
Analyzing periodicity is akin to being a detective looking for clues to predict when events will repeat. For functions, this involves a bit of mathematical investigation. We assess how factors like coefficients and exponents alter the behavior of a function.
Take, for example, the exercise \(\sin(5x)\). To find its periodicity, we consider the influence of the multiplier '5'. It's squeezing our standard sine wave, making its cycle complete 5 times faster. Therefore, the function's values will repeat after a much shorter interval on the x-axis: the new fundamental period is \(\frac{2\pi}{5}\). This process exemplifies how we analyze periodic functions — by examining the impact of mathematical transformations on their fundamental period.
Take, for example, the exercise \(\sin(5x)\). To find its periodicity, we consider the influence of the multiplier '5'. It's squeezing our standard sine wave, making its cycle complete 5 times faster. Therefore, the function's values will repeat after a much shorter interval on the x-axis: the new fundamental period is \(\frac{2\pi}{5}\). This process exemplifies how we analyze periodic functions — by examining the impact of mathematical transformations on their fundamental period.
