Textbook Question
In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan⁻¹ (tan 2π/3)
Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblem 51
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblem 51Chapter 2, Problem 51
In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 2 cos (2πx + 8π)
Verified step by step guidance1
Identify the general form of the cosine function: \(y = A \cos(Bx + C)\), where \(A\) is the amplitude, \(B\) affects the period, and \(C\) affects the phase shift.
Find the amplitude by taking the absolute value of \(A\). In this case, \(A = 2\), so the amplitude is \(|2| = 2\).
Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B = 2\pi\), so substitute to get \(\text{Period} = \frac{2\pi}{2\pi}\).
Determine the phase shift using the formula \(\text{Phase shift} = -\frac{C}{B}\). Given \(C = 8\pi\) and \(B = 2\pi\), substitute to find the phase shift.
To graph one period of the function, start at the phase shift on the x-axis, then mark points at intervals of \(\frac{\text{Period}}{4}\) to capture key points of the cosine wave (maximum, zero crossing, minimum, zero crossing, maximum). Use the amplitude to determine the y-values at these points.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude of a Trigonometric Function
Amplitude is the maximum absolute value of the function's output, representing the height from the midline to the peak of the wave. For functions like y = a cos(bx + c), the amplitude is the absolute value of 'a'. In this example, the amplitude is |2| = 2, indicating the wave oscillates 2 units above and below the midline.
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Introduction to Trigonometric Functions
Period of a Trigonometric Function
The period is the length of one complete cycle of the wave, calculated as (2π) divided by the absolute value of the coefficient 'b' in y = a cos(bx + c). Here, with b = 2π, the period is 2π / 2π = 1. This means the function repeats every 1 unit along the x-axis.
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Phase Shift of a Trigonometric Function
Phase shift is the horizontal translation of the graph, found by solving bx + c = 0 for x, giving x = -c/b. It indicates how far the graph shifts left or right from the origin. For y = 2 cos(2πx + 8π), the phase shift is -8π / 2π = -4, meaning the graph shifts 4 units to the left.
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Related Practice
Textbook Question
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin⁻¹(sin π/3)
Textbook Question
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin⁻¹(cos 2π/3)
Textbook Question
In Exercises 53–54, let f(x) = 2 sec x, g(x) = −2 tan x, and h(x) = 2x − π/2. Graph two periods of y = (f∘h)(x).
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Textbook Question
In Exercises 53–60, use a vertical shift to graph one period of the function. y = sin x + 2
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Textbook Question
In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin π)