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Chapter 10: Problem 38

Solve the given problems.Use a calculator to view the indicated curves.Find the function and graph it for a function of the form \(y=2 \cos (2 x-c)\) that passes through \((\pi / 6,2)\) and for which \(c\) has the smallest possible positive value.

Short Answer

Expert verified
The function is \(y = 2 \cos(2x - \pi/3)\). This graph passes through \((\pi/6, 2)\) with the smallest positive phase shift.

Step by step solution

01

Understand the Given Function

The function is of the form \(y = 2 \cos(2x - c)\). This is a cosine function with an amplitude of 2 and a frequency component represented by \(2x\). The variable \(c\) is the phase shift, and we need to determine its value such that the function passes through the point \((\pi/6, 2)\) with the smallest positive \(c\).
02

Substitute the Point into the Function

Substitute \(x = \pi/6\) and \(y = 2\) into the function: \(2 = 2 \cos(2(\pi/6) - c)\). This simplifies to \(1 = \cos(\pi/3 - c)\).
03

Solve for the Inside of the Cosine

In order for \(\cos(\theta) = 1\), \(\theta\) must equal \(2n\pi\) for any integer \(n\). So, set \(\pi/3 - c = 2n\pi\).
04

Simplify and Solve for \(c\)

Rearrange the equation to solve for \(c\): \(c = \pi/3 - 2n\pi\).
05

Determine the Smallest Positive \(c\)

To find the smallest positive \(c\), set \(n = 0\) to get \(c = \pi/3\). There are no smaller positive values for \(c\) because increasing \(n\) would make \(c\) negative or convert it into a larger positive\/negative integer multiple of \(2\pi\).
06

Verify the Solution

Substitute \(c = \pi/3\) back into the original function to verify: \(y = 2 \cos(2x - \pi/3)\). Insert \(x = \pi/6\) and calculate: \(y = 2 \cos((\pi/3) - \pi/3) = 2 \cos(0) = 2\). The point \((\pi/6, 2)\) satisfies the function.
07

Graph the Function

Graph the function \(y = 2 \cos(2x - \pi/3)\). This function has an amplitude of 2, a frequency of 2 cycles over \(2\pi\), and is shifted \(\pi/3\) to the right.

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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 a fundamental component of trigonometry. It is used to describe oscillations and waves, such as sound and electromagnetic waves. The basic form of a cosine function is written as \(y = a \cos(bx + c) + d\), where each parameter plays a specific role:
  • \(a\) is the amplitude, which determines the height of the wave from the center line to its peak. In our exercise, the amplitude is 2, indicating the wave reaches 2 units above and below the x-axis.
  • \(b\) influences the frequency of the function, which we will explore in another section.
  • \(c\) controls the phase shift, a topic we will discuss in depth soon.
  • \(d\) is the vertical shift, moving the entire function up or down, but is not utilized in our problem.
Understanding the cosine function's parameters allows you to graph and interpret various trigonometric situations. Cosine is particularly symmetric, helping model phenomena with smooth periodic upticks and downturns, making it a staple in mathematics and physics.
Phase Shift
Phase shift in trigonometry refers to the horizontal translation of a function. It determines how far left or right the function is shifted from its standard position. The phase shift is calculated from the expression \(bx + c\) in the cosine function \(y = a \cos(bx + c)\).
  • When the phase shift is positive, the graph of the function shifts to the left. Conversely, a negative phase shift moves the graph to the right.
  • In our problem, the function is expressed as \(y = 2 \cos(2x - c)\), where the phase shift \(c\) is what we needed to calculate. The smallest positive value for \(c\) we found was \(\pi/3\), meaning the function shifts \(\pi/3\) units to the right.
  • The phase shift is essential for positioning the wave correctly to match boundary conditions or specific data points, like we did with the point \((\pi / 6, 2)\).
Understanding phase shifts helps in aligning waves and functions in practical applications like signal processing and acoustics.
Frequency in Trigonometry
The frequency of a trigonometric function refers to how often the function's cycle repeats over a particular interval. In the context of the cosine function \(y = a \cos(bx + c)\), frequency is controlled by the parameter \(b\).
  • A higher frequency indicates more cycles over a given period. For a function \(y = 2 \cos(2x - c)\), such as in our problem, the coefficient of \(x\) (which is 2) indicates the frequency. This means that there are 2 complete cycles of the cosine wave every \(2\pi\) units.
  • In trigonometric terms, frequency is often represented as the number of periods or oscillations within the interval \([0, 2\pi]\).
  • Frequency is an important concept when analyzing waves and oscillatory behavior in both natural and technological contexts. It explains how rapidly phenomena like sound waves, alternating currents, or vibrations occur.
Mastering frequency allows students to predict and manipulate the behavior of waves and responding systems accurately.

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Most popular questions from this chapter

Show that the given equations are correct. The method using a calculator is indicated in Exercise 31 Show that \(\sin (x / 2-3 \pi / 4)=-\sin (3 \pi / 4-x / 2)\)

In Exercises \(47-50,\) graph the given functions. In Exercises 47 and \(48,\) first rewrite the function with a positive angle, and then graph the resulting function. $$y=3 \sin (-2 x)$$

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Show that the given equations are correct. The method using a calculator is indicated in Exercise 31 Show that \(\cos (2 x-3 \pi / 8)=\cos (3 \pi / 8-2 x)\)

View at least two cycles of the graphs of the given functions on a calculator. $$y=18 \csc \left(3 x-\frac{\pi}{3}\right)$$
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