Domanda del libro di testo
In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan⁻¹ (tan 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 51
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 51Capitolo 2, Problema 51
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin⁻¹(cos 2π/3)
Guida verificata passo dopo passo1
Recall that the function \( \sin^{-1}(x) \) (also called arcsine) gives the angle \( \theta \) in the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) such that \( \sin(\theta) = x \).
First, evaluate \( \cos \frac{2\pi}{3} \). Use the unit circle or cosine properties to find the exact value of \( \cos \frac{2\pi}{3} \).
Once you have \( \cos \frac{2\pi}{3} = x \), rewrite the original expression as \( \sin^{-1}(x) \).
Next, find the angle \( \theta \) in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) such that \( \sin(\theta) = x \). This may involve using the identity \( \sin(\theta) = \sin(\pi - \theta) \) or considering the symmetry of sine and cosine functions.
Express the final answer as the exact angle \( \theta \) in radians that satisfies the above conditions, without using a calculator.
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Inverse Sine Function (sin⁻¹ or arcsin)
The inverse sine function, sin⁻¹(x), returns the angle whose sine is x, with a principal range of [-π/2, π/2]. It is used to find angles from known sine values, and understanding its range is crucial to correctly interpreting results.
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Percorso guidato
4:03
Inverse Sine
Evaluating Cosine at Special Angles
Cosine values at special angles like 2π/3 are well-known and can be found using the unit circle. For 2π/3, cos(2π/3) equals -1/2. Recognizing these values helps simplify expressions before applying inverse trigonometric functions.
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Percorso guidato
3:48Evaluate Composite Functions - Special Cases
Relationship Between Trigonometric Functions and Their Inverses
When evaluating expressions like sin⁻¹(cos θ), it is important to understand how the cosine value fits within the domain and range of the inverse sine function. This often involves converting cosine values to sine values or considering angle identities to find the correct angle.
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Percorso guidato
4:28Introduction to Inverse Trig Functions
Pratica correlata
Domanda del libro di testo
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin⁻¹(sin π/3)
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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 45–52, graph two periods of each function. y = csc|x|
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Domanda del libro di testo
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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