Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
4. Graphing Trigonometric Functions
Graphs of the Sine and Cosine Functions
Problem 4.15Lial - 12th Edition
Textbook Question
Textbook QuestionGraph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.
y = ⅔ sin x
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude
Amplitude refers to the maximum distance a wave reaches from its central position, which is crucial in understanding the height of trigonometric functions like sine and cosine. For the function y = ⅔ sin x, the amplitude is ⅔, indicating that the graph will oscillate between -⅔ and ⅔. This concept helps in visualizing the vertical stretch or compression of the sine wave.
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Graphing Trigonometric Functions
Graphing trigonometric functions involves plotting the values of the function over a specified interval, in this case, [-2π, 2π]. Understanding the periodic nature of sine functions, which repeat every 2π, is essential for accurately representing the function's behavior across the given interval. This includes identifying key points such as intercepts, maximums, and minimums.
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Period of the Sine Function
The period of a sine function is the length of one complete cycle of the wave. For the standard sine function, the period is 2π, meaning it repeats every 2π units along the x-axis. In the function y = ⅔ sin x, the period remains 2π, which is important for determining how many cycles will fit within the interval [-2π, 2π] and for accurately sketching the graph.
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Related Practice
Textbook Question
In Exercises 1–6, determine the amplitude of each function. Then graph the function and y = sin x in the same rectangular coordinate system for 0 ≤ x ≤ 2π.
y = 4 sin x
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Graphs of the Sine and Cosine Functions practice set
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