Domanda del libro di testo
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sec⁻¹ (−1)
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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 37
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 37Capitolo 2, Problema 37
In Exercises 29–44, graph two periods of the given cosecant or secant function. y = −2 csc πx
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Identify the given function: \(y = -2 \csc(\pi x)\). Recall that \(\csc(\theta) = \frac{1}{\sin(\theta)}\), so this function is related to the sine function.
Determine the period of the function. The general period of \(\csc(bx)\) is \(\frac{2\pi}{b}\). Here, \(b = \pi\), so the period is \(\frac{2\pi}{\pi} = 2\).
Since the problem asks for two periods, calculate the interval for \(x\) over which to graph: from \(0\) to \(4\) (because one period is 2, two periods is \(2 \times 2 = 4\)).
Find the key points where the sine function (and thus the cosecant function) is zero, because \(\csc(\theta)\) is undefined where \(\sin(\theta) = 0\). Solve \(\sin(\pi x) = 0\) for \(x\) in \([0,4]\), which occurs at integer values \(x = 0, 1, 2, 3, 4\).
Plot the vertical asymptotes at these points where the function is undefined. Then, plot the shape of \(y = -2 \csc(\pi x)\) between these asymptotes, remembering that the negative sign reflects the graph over the x-axis and the amplitude 2 stretches it vertically.
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Understanding the Cosecant Function
The cosecant function, csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). It is undefined where sin(x) = 0, leading to vertical asymptotes. Its graph consists of branches that approach these asymptotes and have minimum or maximum points where sine reaches ±1.
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Percorso guidato
6:22
Graphs of Secant and Cosecant Functions
Effect of Transformations on Trigonometric Graphs
Multiplying the cosecant function by a constant, such as -2, affects its amplitude and orientation. The factor 2 stretches the graph vertically, making peaks and troughs twice as far from the x-axis, while the negative sign reflects the graph across the x-axis, inverting its shape.
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Percorso guidato
5:25Introduction to Transformations
Period of the Cosecant Function with Horizontal Scaling
The period of csc(x) is 2π, but when the input is scaled by a factor (like πx), the period changes. The period is calculated as 2π divided by the coefficient of x inside the function. For y = -2 csc(πx), the period is 2π/π = 2, so two periods span an interval of length 4 on the x-axis.
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Percorso guidato
6:22Graphs of Secant and Cosecant Functions
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In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = 4 cos 2πx
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In Exercises 37–40, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, graph one period of the equation. Then find the following: a. the maximum displacement b. the frequency c. the time required for one cycle d. the phase shift of the motion. d = 3 cos(πt + π/2)
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Domanda del libro di testo
In Exercises 29–44, graph two periods of the given cosecant or secant function. y = −1/2 sec πx
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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 0.9)
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Domanda del libro di testo
In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. ___ tan⁻¹ (−√473)
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