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Chapter 4: Problem 11

In Exercises 11-24, state the amplitude and period of each sinusoidal function. $$ y=\frac{3}{2} \cos (3 x) $$

Short Answer

Expert verified
The amplitude is \( \frac{3}{2} \) and the period is \( \frac{2\pi}{3} \).

Step by step solution

01

Identify the Sinusoidal Function Parameters

The given function is of the form \( y = a \cdot \cos(bx) \), where \( a \) represents the amplitude and \( b \) affects the period. In this exercise, \( a = \frac{3}{2} \) and \( b = 3 \).
02

Determine the Amplitude

The amplitude of a sinusoidal function \( y = a \cdot \cos(bx) \) is the absolute value of the coefficient \( a \). Thus, the amplitude is \( |\frac{3}{2}| = \frac{3}{2} \).
03

Calculate the Period

The period of a cosine function \( y = a \cdot \cos(bx) \) is given by \( \frac{2\pi}{b} \). For this function, with \( b = 3 \), the period is \( \frac{2\pi}{3} \).

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Key Concepts

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

Trigonometric Functions
Trigonometric functions are a fundamental part of mathematics that study the relationships between angles and lengths in triangles.
There are three primary trigonometric functions: sine, cosine, and tangent.
  • Sine (sin) relates the opposite side over the hypotenuse in a right triangle.
  • Cosine (cos) relates the adjacent side over the hypotenuse.
  • Tangent (tan) relates the opposite side over the adjacent side.
These functions are periodic, meaning they repeat their values in regular intervals. Understanding these functions is vital, as they form the basis of many concepts in mathematics and science, including waves and signal processing.
Cosine Function
The cosine function is one of the primary trigonometric functions and it is often represented as \( y = rac{3}{2} \cos(3x) \). In this function, the cosine wave starts at its maximum value and oscillates between the amplitude values.
The general equation of a cosine function is given by: \( y = a \cos(bx + c) + d \), where
  • \(a\) represents the amplitude
  • \(b\) controls the period
  • \(c\) affects the phase shift
  • \(d\) represents the vertical shift
The cosine function is particularly notable for its symmetry and its ability to model oscillating behaviors and patterns.
Sinusoidal Function Parameters
Sinusoidal functions can be identified by their parameters in the equation \( y = a \cos(bx + c) + d \). Each parameter plays a critical role in the shape and position of the wave:
  • Amplitude (\(a\)): Determines the height of the peaks and depths of the troughs from the central line.
  • Frequency (\(b\)): Inversely affects the period of the wave. Higher values of \(b\) make the wave cycle more frequently.
  • Phase Shift (\(c\)): Moves the wave left or right along the x-axis.
  • Vertical Shift (\(d\)): Moves the wave up or down along the y-axis.
These parameters can be effortlessly manipulated to match various practical applications like sound waves, electromagnetic waves, and alternating currents.
Amplitude
The amplitude of a sinusoidal function, notably a cosine function, refers to the maximum distance from its central axis or mean position. For \( y = \frac{3}{2} \cos(3x) \),
  • The amplitude is determined by the coefficient in front of the cosine, in this case, \(\frac{3}{2}\).
  • The amplitude represents the spread or intensity of the wave.
  • It is calculated as the absolute value, \(|a|\), ensuring it is always a positive quantity.
Being aware of the amplitude gives insight into the range of motion or oscillation of various real-world scenarios modeled by sinusoidal functions.
Period Calculation
The period of a sinusoidal function indicates how long it takes to complete one full cycle of the wave. For a cosine function \( y = a \cos(bx) \), the period is calculated using the formula \[ \text{Period} = \frac{2\pi}{|b|} \].
For the given function \( y = \frac{3}{2} \cos(3x) \),
  • Set \(b = 3\),
  • the period would be \(\frac{2\pi}{3}\).
Understanding period calculation is essential, especially when we model time-dependent phenomena such as tides, sound waves, and light waves. This concept tells us how frequently these phenomena repeat their behavior over a consistent interval.

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Most popular questions from this chapter

In Exercises 31-42, graph the functions over the indicated intervals. $$ y=-4 \csc (x+\pi), \text { over at least one period } $$

In Exercises 31-42, graph the functions over the indicated intervals. $$ y=2 \csc (2 x+\pi),-2 \pi \leq x \leq 2 \pi $$

In Exercises 51-62, add the ordinates of the indicated functions to graph each summed function on the indicated interval. $$ y=\sin \left(\frac{x}{2}\right)+\sin (2 x), \text { on } 0 \leq x \leq 4 \pi $$

In Exercises 31-42, graph the functions over the indicated intervals. $$ y=-\sec (2 \pi x),-1 \leq x \leq 1 $$

Solve the equation \(\tan (2 x-\pi)=0\) for \(x\) in the interval \([-\pi, \pi]\) by graphing.
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