/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Sketch one cycle for the followi... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 7

Sketch one cycle for the following graphs starting with \(t=0\). Show important points. a \(7 \sin (3 t)\) b \(2 \cos \left(\frac{t}{2}\right)\) c \(3 \sin \left(t+\frac{\pi}{3}\right)\)

Short Answer

Expert verified
Graph a): Period \( \frac{2\pi}{3} \). Graph b): Period \( 4\pi \). Graph c): Period \( 2\pi \), Phase shift \( -\frac{\pi}{3} \).

Step by step solution

01

- Understanding the Function

For each function, recognize the type of trigonometric function (sine or cosine), amplitude, period, and phase shift. This helps identify important points to sketch.
02

- Analyze and Sketch: Graph a)

A cycle of the function \(7 \sin (3t)\) can be determined by identifying its components:- **Amplitude**: 7- **Period**: The period of \( \sin (kt) \) is \( \frac{2\pi}{k} \). Here, \(k = 3\). Thus, the period is \( \frac{2\pi}{3} \).- **Important points**: Start from \( t = 0 \) and include points where the sine function has its maximum, minimum, and zeros within one period.Sketch the graph from \( t = 0 \) to \( t = \frac{2\pi}{3} \), marking the maximum at \( t = \frac{\pi}{6} \), minimum at \( t = \frac{\pi}{2} \), and return to zero at \( t = \frac{2\pi}{3} \).
03

- Analyze and Sketch: Graph b)

A cycle of the function \(2 \cos \left(\frac{t}{2}\right)\) can be determined by identifying its components:- **Amplitude**: 2- **Period**: The period of \( \cos \left(\frac{t}{2}\right) \) is \(2\pi \times 2 = 4\pi \).- **Important points**: Start from \( t = 0 \) and include points where the cosine function has its maximum, minimum, and zeros within one period.Sketch the graph from \( t = 0 \) to \( t = 4\pi \), marking the maximum at \( t = 0 \) and \( t = 4\pi \), the minimum at \( t = 2\pi \), and the zeros at \( t = \pi \) and \( t = 3\pi \).
04

- Analyze and Sketch: Graph c)

A cycle of the function \(3 \sin \left(t+\frac{\pi}{3}\right)\) can be determined by identifying its components:- **Amplitude**: 3- **Period**: The period of \( \sin (t + \text{constant}) \) is unchanged at \(2\pi\).- **Phase Shift**: Since there is a phase shift of \(\frac{\pi}{3}\), the graph starts at \(t = -\frac{\pi}{3}\).Sketch the graph from \( t = -\frac{\pi}{3} \) to \( t = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \), marking maximum, minimum, and zeros appropriately.

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Key Concepts

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

sine function
The sine function is a fundamental trigonometric function that describes a smooth, periodic oscillation. In general, the sine function is denoted as \(\text{sin}(t)\), where \(t\) represents the angle or time. The sine function has important characteristics:
  • The graph of the sine function is a wave-like pattern that repeats every \(2\pi\).
  • It starts at zero, rises to a maximum value of 1, returns to zero, falls to a minimum value of -1, and then returns to zero to complete one cycle.
  • This pattern continues indefinitely in both directions along the t-axis.
In the context of the exercise, understanding the sine function helps identify key points and shapes when sketching graphs of modified sine functions, such as \(7 \sin(3t)\).
cosine function
The cosine function is another essential trigonometric function known for its periodic oscillations. It is denoted as \(\text{cos}(t)\), where \(t\) represents the angle or time. Key points about the cosine function include:
  • Like the sine function, the cosine function has a periodic pattern that repeats every \(2\pi\).
  • It starts at its maximum value of 1 when \(t = 0\), then decreases to zero, reaches a minimum value of -1, returns to zero, and finally goes back to its starting maximum value to complete one cycle.
For example, in problem \(\text{b})\), the function \(2 \cos\bigg(\frac{t}{2}\bigg)\) uses the basic cosine shape but with adjustments to its amplitude and period. By understanding these modifications, we can sketch the graph accurately.
amplitude
Amplitude is a measure of how much a trigonometric function oscillates above and below its central value. For the sine and cosine functions, amplitude affects the height of the peaks and the depths of the troughs.
  • Mathematically, amplitude is the coefficient in front of the sine or cosine function. For example, in the function \(7 \sin(3t)\), the amplitude is 7.
  • This means that the graph of this function will oscillate between +7 and -7 instead of the default +1 and -1.
Recognizing the amplitude helps in marking the maximum and minimum points while sketching the function, which is vital for creating an accurate graph.
period
The period of a trigonometric function is the distance along the t-axis before the function starts repeating itself. For the standard sine and cosine functions, the period is \(2\pi\). However, this period can change if the function is modified.
  • For example, in the function \(7 \sin(3t)\), the factor of 3 affects the period of the sine function.
  • The new period is calculated using the formula \(\frac{2\pi}{k}\), where \(k\) is the coefficient of \(t\). Here, the period is \(\frac{2\pi}{3}\).
In general, understanding the period of a function allows us to know when to start and stop the sketch for one complete cycle, ensuring the graph is properly scaled on the t-axis.
phase shift
Phase shift refers to the horizontal movement of a trigonometric graph along the t-axis. This occurs when there is a constant added or subtracted inside the function's argument.
  • For instance, in the function \(\text{sin}(t + \frac{\text{\text{蟺}}}{3})\), the phase shift is \(\frac{\text{\text{蟺}}}{3}\) to the left.
  • This shift means that instead of starting the sine wave at t=0, it starts at \(t=-\frac{\text{\text{蟺}}}{3}\).
Recognizing the phase shift helps in identifying the correct starting point for the graph and correctly aligning the key points (e.g., maximum, minimum, and zero crossings) along the t-axis. This concept ensures the accuracy of the graph's position on the horizontal axis.

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Most popular questions from this chapter

Find all the unknown sides and angles of a triangle \(\mathrm{ABC}\) for the following: a \(\mathrm{AC}=10, \mathrm{BC}=6\) and angle \(A=30^{\circ}\) b \(\mathrm{AB}=7, \mathrm{BC}=9\) and angle \(C=35^{\circ}\).

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