Domanda del libro di testo
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sec⁻¹ (−1)
757
views
Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
Tutti i libri di testo
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 39
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 39Capitolo 2, Problema 39
In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin(sin⁻¹ 0.9)
Guida verificata passo dopo passo1
Recognize that the expression is \( \sin(\sin^{-1}(0.9)) \). The function \( \sin^{-1} \) (also called arcsin) is the inverse of the sine function, which means it 'undoes' the sine function within its domain.
Recall the property of inverse functions: for any value \( x \) in the domain of \( \sin^{-1} \), \( \sin(\sin^{-1}(x)) = x \). This holds true because \( \sin^{-1}(x) \) gives the angle whose sine is \( x \).
Check that the input value \( 0.9 \) is within the domain of \( \sin^{-1} \), which is \( [-1, 1] \). Since \( 0.9 \) is within this range, the property applies directly.
Therefore, the expression simplifies directly to \( 0.9 \) without further calculation.
Summarize that \( \sin(\sin^{-1}(0.9)) = 0.9 \) because the sine and arcsine functions are inverses on the domain \( [-1, 1] \).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
1mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Inverse Sine Function (sin⁻¹ or arcsin)
The inverse sine function, denoted sin⁻¹ or arcsin, returns the angle whose sine is a given value. It is defined for inputs between -1 and 1 and outputs angles in the range [-π/2, π/2]. Understanding this helps in interpreting expressions like sin(sin⁻¹ x).
Video consigliato:
Percorso guidato
4:03
Inverse Sine
Composition of Sine and Inverse Sine Functions
When sine and its inverse are composed as sin(sin⁻¹ x), the result simplifies to x for all x in the domain [-1, 1]. This property is crucial for evaluating expressions without a calculator, as it allows direct simplification.
Video consigliato:
Percorso guidato
4:03Inverse Sine
Domain and Range Restrictions
The domain of sin⁻¹ is [-1, 1], and its range is limited to angles between -π/2 and π/2. Recognizing these restrictions ensures the expression is valid and helps avoid errors when simplifying or evaluating trigonometric expressions.
Video consigliato:
Percorso guidato
4:22Domain and Range of Function Transformations
Pratica correlata
Domanda del libro di testo
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)
901
views
Domanda del libro di testo
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ cos(sin⁻¹ √2/2)
729
views
Domanda del libro di testo
In Exercises 29–44, graph two periods of the given cosecant or secant function. y = −1/2 sec πx
659
views
Domanda del libro di testo
In Exercises 29–44, graph two periods of the given cosecant or secant function. y = −2 csc πx
720
views
Domanda del libro di testo
In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = -4 cos 1/2 x
560
views