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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 13
Capitolo 7, Problema 13

Find the exact value of each real number y if it exists. Do not use a calculator.
y = sin⁻¹ 0

Guida verificata passo dopo passo
1
Understand that the problem asks for the values of \(y\) such that \(\sin y = 0\), where \(y = \sin^{-1} 0\) represents the inverse sine (arcsine) function.
Recall the definition of the inverse sine function: \(y = \sin^{-1} x\) means \(\sin y = x\) and \(y\) lies within the principal range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
Set up the equation based on the problem: \(\sin y = 0\) with \(y \in [-\frac{\pi}{2}, \frac{\pi}{2}]\).
Identify all angles \(y\) in the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\) where \(\sin y = 0\). These are the values where the sine function crosses zero within the principal range.
Conclude that the exact value(s) of \(y\) satisfying the equation are those angles found in the previous step, which represent the output of \(\sin^{-1} 0\).

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Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function, denoted as sin⁻¹ or arcsin, returns the angle whose sine is a given number. It is defined for inputs between -1 and 1 and outputs angles in the range [-π/2, π/2]. Understanding this function helps find angles corresponding to specific sine values.
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Inverse Sine

Sine Function Values

The sine function relates an angle to the ratio of the opposite side over the hypotenuse in a right triangle. Key values include sin(0) = 0, sin(π/2) = 1, and sin(-π/2) = -1. Recognizing these standard values aids in identifying angles for given sine outputs.
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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Domain and Range Restrictions of Inverse Trigonometric Functions

Inverse trigonometric functions have restricted ranges to ensure they are functions. For arcsin, the output is limited to angles between -π/2 and π/2. This restriction means that for sin⁻¹(0), the principal value is 0, even though sine of other angles like π also equals zero.
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Domain and Range of Function Transformations
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