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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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 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin(sin⁻¹ π)
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Recognize that the expression is \( \sin(\sin^{-1} \pi) \). The function \( \sin^{-1} \) (also called arcsin) is the inverse sine function, which returns an angle whose sine is the given value.
Recall the domain and range of the inverse sine function: \( \sin^{-1} x \) is defined only for \( x \) in the interval \( [-1, 1] \), and it returns an angle \( \theta \) in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).
Since \( \pi \approx 3.14159 \) is outside the domain \( [-1, 1] \) of \( \sin^{-1} \), the expression \( \sin^{-1} \pi \) is not defined in the real numbers.
Therefore, the expression \( \sin(\sin^{-1} \pi) \) cannot be evaluated as a real number because \( \sin^{-1} \pi \) does not exist in the real domain.
Conclude that the exact value of \( \sin(\sin^{-1} \pi) \) is undefined or does not exist in the real number system.
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Inverse Sine Function (sin⁻¹ or arcsin)
The inverse sine function, denoted sin⁻¹ or arcsin, returns the angle whose sine is a given number. Its output range is limited to [-π/2, π/2], meaning it only accepts inputs between -1 and 1 for real values.
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4:03
Inverse Sine
Domain Restrictions of the Sine Function
The sine function outputs values between -1 and 1, so its inverse function sin⁻¹ only accepts inputs within this range. If the input to sin⁻¹ is outside [-1, 1], the expression is undefined in the real number system.
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3:43Finding the Domain of an Equation
Evaluating Composite Trigonometric Expressions
When evaluating expressions like sin(sin⁻¹ x), if x is within the domain of sin⁻¹, the result simplifies to x. However, if x is outside the domain, the expression cannot be evaluated without extending to complex numbers.
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3:48Evaluate Composite Functions - Special Cases
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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 54–57, solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. A = 22.3°, c = 10
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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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In Exercises 53–60, use a vertical shift to graph one period of the function. y = sin x + 2
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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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