Domanda del libro di testo
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin⁻¹(cos 2π/3)
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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 FunctionsProblema 53
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 53Capitolo 2, Problema 53
In Exercises 53–60, use a vertical shift to graph one period of the function. y = sin x + 2
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Identify the base function and its characteristics. Here, the base function is \(y = \sin x\), which has a period of \(2\pi\), an amplitude of 1, and oscillates between -1 and 1.
Understand the effect of the vertical shift. The function \(y = \sin x + 2\) shifts the entire sine curve upward by 2 units, so the midline of the graph moves from \(y=0\) to \(y=2\).
Determine the new range of the function after the vertical shift. Since the original sine function ranges from -1 to 1, adding 2 shifts this range to \(1 \leq y \leq 3\).
Plot one period of the function from \(x=0\) to \(x=2\pi\). At key points: \(x=0\), \(x=\frac{\pi}{2}\), \(x=\pi\), \(x=\frac{3\pi}{2}\), and \(x=2\pi\), calculate the corresponding \(y\) values using \(y = \sin x + 2\).
Sketch the graph using the calculated points, ensuring the wave oscillates smoothly between \(y=1\) and \(y=3\) with the midline at \(y=2\), completing one full period over the interval \([0, 2\pi]\).
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Vertical Shift in Trigonometric Functions
A vertical shift moves the entire graph of a function up or down without changing its shape. For y = sin x + 2, the '+2' shifts the sine curve two units upward, raising the midline from y = 0 to y = 2.
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6:31
Phase Shifts
Period of the Sine Function
The period of the sine function is the length of one complete cycle, which is 2π for y = sin x. Graphing one period means plotting the function from 0 to 2π to capture a full wave of the sine curve.
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5:33Period of Sine and Cosine Functions
Graphing Sine Functions
Graphing sine functions involves plotting key points such as maxima, minima, and zeros over one period. For y = sin x + 2, these points are shifted vertically, so the maximum is at y = 3, minimum at y = 1, and midline at y = 2.
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5:53Graph of Sine and Cosine Function
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Domanda del libro di testo
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)
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Domanda del libro di testo
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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Domanda del libro di testo
In Exercises 52–53, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. sec(sin⁻¹ 1/x)
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Domanda del libro di testo
In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin(sin⁻¹ π)
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Domanda del libro di testo
In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin π)
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