Textbook Question
An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.
𝒮(t) = 5 cos 2t
What is the frequency?
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Ch. 4 - Graphs of the Circular Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 5, Problem 1
Fill in the blank(s) to correctly complete each sentence.
The graph of y = sin (x + π/4) is obtained by shifting the graph of y = sin x ______ unit(s) to the ________ (right/left).
Verified step by step guidance1
Recall that the function y = sin(x + c) represents a horizontal shift of the basic sine function y = sin x, where c is a constant.
If the function is y = sin(x + \(\frac{\pi}{4}\)), the graph is shifted horizontally by \(\frac{\pi}{4}\) units.
Since the argument inside the sine function is (x + \(\frac{\pi}{4}\)), this corresponds to a shift to the left by \(\frac{\pi}{4}\) units (because adding inside the function shifts the graph left).
Therefore, the graph of y = sin(x + \(\frac{\pi}{4}\)) is obtained by shifting the graph of y = sin x \(\frac{\pi}{4}\) unit(s) to the left.
This is a standard property of function transformations: y = sin(x + c) shifts the graph left by c units, and y = sin(x - c) shifts it right by c units.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Phase Shift in Trigonometric Functions
Phase shift refers to the horizontal translation of a trigonometric graph caused by adding or subtracting a constant inside the function's argument. For y = sin(x + π/4), the graph shifts horizontally by π/4 units. The sign inside the function determines the direction of the shift.
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6:31
Phase Shifts
Graph of the Sine Function
The sine function y = sin x is periodic with period 2π and oscillates between -1 and 1. Understanding its basic shape and key points helps in visualizing transformations like shifts, stretches, and reflections applied to the graph.
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5:53Graph of Sine and Cosine Function
Effect of Adding a Positive Constant Inside the Function Argument
Adding a positive constant inside the argument of sine, as in sin(x + c), shifts the graph to the left by c units. This is because the input values that produce the same output occur earlier on the x-axis, effectively moving the graph leftward.
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3:30Example 3
Related Practice
Textbook Question
Fill in the blank(s) to correctly complete each sentence.
The graph of y = cos (x - π/6) is obtained by shifting the graph of y = cos x ______ unit(s) to the ________ (right/left).
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Textbook Question
An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.
𝒮(t) = 5 cos 2t
What is the amplitude of this motion?
700
views
Textbook Question
An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.
𝒮(t) = 5 cos 2t
What is the period of this motion?
647
views