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Chapter 5: Problem 3

The period of a sine or cosine function is given by _________.

Short Answer

Expert verified
The period of a sine or cosine function is given by \(\frac{2\pi}{|b|}\), where 'b' is the coefficient of x in the function equation.

Step by step solution

01

Understand the concept of a period

A period of a function refers to the interval over which the function's values repeat. In terms of graph, it is the horizontal length of the shortest part of the graph that consists of the entire pattern. For sine or cosine function, a period is the length of one complete wave or cycle.
02

Recognize relation to 2蟺

The period of standard sine and cosine function (y = sin(x) or y = cos(x)) is \(2\pi\), as these functions repeat their values every \(2\pi\) radians.
03

Understand the effect of coefficient on the period

In general, for any function with form \(y = a\sin(bx)\) or \(y = a\cos(bx)\), the 'b' indicates a horizontal stretch or compression relative to the standard function. The effect of 'b' modifies the standard period \(2\pi\), given the formula for period as \(\frac{2\pi}{|b|}\). So, the absolute value of 'b' is the number of periods that fit into the \(2\pi\) interval.

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Key Concepts

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

Periodicity
Periodicity in trigonometric functions is a fascinating concept that involves the repeating nature of these functions over a specific interval. A function is periodic if it repeats its values in regular intervals or periods. For example, when we talk about the periodicity of sine or cosine functions, we mean that their patterns repeat every full cycle. This characteristic makes trigonometric functions predictable and is utilized in various applications from signal processing to time series analysis.
  • The period of a sine or cosine function is the length of the smallest interval for which the function repeats.
  • A standard sine or cosine function, such as \(y = \sin(x)\) or \(y = \cos(x)\), repeats every \(2\pi\) radians, which is just over 6.28 radians.
  • Alterations in the function, such as changing the coefficient of \(x\), will modify this period, compressing or stretching the cycle length.
Understanding periodicity is essential when working with repetitive patterns and waves, providing insights into how these functions behave over their intervals.
Sine Function
The sine function is one of the fundamental trigonometric functions, known for its smooth and continuous wave-like pattern. It is often represented as \(y = \sin(x)\), and this basic form has a standard period of \(2\pi\). This means that the sine wave repeats its complete cycle, going from zero to one, to zero, to minus one, and back to zero again every \(2\pi\) units on the x-axis.
  • At the beginning of its cycle, \(\sin(x) = 0\) at \(x = 0\).
  • It reaches its maximum value of 1 at \(x = \frac{\pi}{2}\).
  • The function then returns to zero at \(x = \pi\), reaches its minimum value of -1 at \(x = \frac{3\pi}{2}\), and completes the cycle back at zero at \(x = 2\pi\).
Sine functions are crucial in modeling periodic phenomena such as sound waves, light waves, and tides, due to their rhythmic oscillation characteristics.
Cosine Function
The cosine function is another essential trigonometric function, similar in properties to the sine function but with a distinct starting point. Expressed as \(y = \cos(x)\), it also has a standard period of \(2\pi\). A cosine wave exhibits a complete oscillation in this interval but starts at its maximum value.
  • The function begins at its peak with \(\cos(x) = 1\) when \(x = 0\).
  • It decreases to zero at \(x = \frac{\pi}{2}\), reaches its minimum value of -1 at \(x = \pi\), and returns to zero at \(x = \frac{3\pi}{2}\).
  • Finally, \(\cos(x)\) returns to 1 at \(x = 2\pi\), completing the cycle.
Cosine functions are widely used in engineering, physics, and various fields where wave forms and oscillation patterns need representation. Their unique waveform characteristics ensure they are pivotal in analyzing cyclical patterns.

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