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Chapter 4: Problem 28

In Exercises 25-40, graph the given sinusoidal functions over one period. $$ y=\cos (3 x) $$

Short Answer

Expert verified
The function \( y = \\cos(3x) \) has a period of \( \frac{2\pi}{3} \) and key points at \( x = 0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3} \), forming one complete wave from \( y = 1 \) to \( y = -1 \) and back to \( y = 1 \).

Step by step solution

01

Identify Key Components of the Function

The given function is \( y = \cos(3x) \). It is a cosine function with a frequency multiplier. The number \( 3 \) attached to \( x \) indicates the frequency. The amplitude is 1, as there is no coefficient before \( \cos \). The cosine function usually has an amplitude (maximum) of 1 and a minimum of -1.
02

Determine the Period of the Function

The period of a cosine function \( y = \cos(bx) \) is given by \( \frac{2\pi}{|b|} \). Since \( b=3 \), the period is \( \frac{2\pi}{3} \). This is the length of one complete cycle of the cosine wave.
03

Plot the Key Points of the Function

Cosine functions have key points at every quarter of their period: \( 0, \frac{\text{period}}{4}, \frac{\text{period}}{2}, \frac{3\times\text{period}}{4}, \) and \( \text{period} \). For \( y = \cos(3x) \), these points are: \( 0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3} \). Evaluate \( y = \cos(3x) \) at these points.
04

Evaluate the Function at Key Points

Calculate \( y = \cos(3x) \) at each key point:- \( x = 0: \cos(0) = 1 \)- \( x = \frac{\pi}{6}: \cos(\frac{\pi}{2}) = 0 \)- \( x = \frac{\pi}{3}: \cos(\pi) = -1 \)- \( x = \frac{\pi}{2}: \cos(\frac{3\pi}{2}) = 0 \)- \( x = \frac{2\pi}{3}: \cos(2\pi) = 1 \)
05

Sketch the Graph Over One Period

Begin plotting at \( x = 0 \) with \( y = 1 \). As you move to each subsequent key point from \( \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2} \) to \( \frac{2\pi}{3} \), plot the points \( 0, -1, 0, \) and finally \( 1 \). Connect these points smoothly to form the typical wave shape of the cosine function.

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Key Concepts

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

Cosine Function
The cosine function is one of the basic trigonometric functions, often noted as \( \cos(x) \). It is widely used in mathematics to represent oscillatory behaviour. The function itself oscillates between values \( -1 \) and \( 1 \), making it perfect for modeling cycles, such as waves or the rotation of a circle.

Here are a few important aspects of the cosine function:
  • The base graph of \( y = \cos(x) \) starts at \( (0, 1) \), dips to \( (\pi, -1) \) at the trough, and returns to \( (2\pi, 1) \) at the peak. This completes one full cycle.
  • In a transformational context, if you find an expression like \( y = \cos(bx) \), the graph will be horizontally stretched or compressed based on the value of \( b \).
  • Standard features such as symmetry, evenness in regard to the y-axis, and periodicity remain key aspects of the cosine function.
Frequency and Period
When dealing with sinusoidal functions like the cosine function, frequency and period are crucial concepts.

**Frequency** refers to how many cycles of the wave occur in a given unit of time. In the function \( y = \cos(bx) \), the coefficient \( b \) alters the frequency.

**Period** is the duration it takes for the wave to complete one full cycle. It is given by \( \frac{2\pi}{|b|} \) for the general cosine function. For example, in the function \( y = \cos(3x) \), the period is \( \frac{2\pi}{3} \), meaning the wave completes a cycle faster than the base function \( y = \cos(x) \).
  • The period determines the "length" of one wavelength on the x-axis.
  • Higher frequency results in a shorter period, causing the waves to appear more frequently within the same span of the x-axis.
Graphing Trigonometric Functions
Graphing trigonometric functions, such as the cosine function, is a methodical process. It involves understanding the structure and transformations of the function.

For \( y = \cos(3x) \), the key steps involve:
  • Identifying key points at regular intervals along the wave period. These intervals are at \( 0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \) and \( \frac{2\pi}{3} \).
  • Calculating and plotting the function values at these points. Notably, these will be \( 1, 0, -1, 0, \) and \( 1 \) respectively.
  • Connecting the plotted points smoothly to illustrate the oscillatory nature of a wave.
Understanding how to plot such functions not only helps in visualizing but also solidifies understanding of periodic behaviours, cycles, and transformations.

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Most popular questions from this chapter

In Exercises 9-18, determine the period and phase shift (if there is one) for each function. $$ y=\sec \left(\frac{1}{2} x+\frac{\pi}{4}\right) $$

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