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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 6.3.27
Capitolo 7, Problema 6.3.27

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
sin x = sin 2x

Guida verificata passo dopo passo
1
Start with the given equation: \(\sin x = \sin 2x\).
Recall the identity that if \(\sin A = \sin B\), then the solutions are given by two cases: \(A = B + 2k\pi\) or \(A = \pi - B + 2k\pi\), where \(k\) is any integer.
Apply this identity to the equation by setting \(x = 2x + 2k\pi\) and \(x = \pi - 2x + 2k\pi\) separately.
Solve each resulting equation for \(x\) within the interval \([0, 2\pi)\) by isolating \(x\) and considering integer values of \(k\) that keep \(x\) in the interval.
List all values of \(x\) found from both cases that lie within \([0, 2\pi)\) as the exact solutions.

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Trigonometric Equation Solving

Solving trigonometric equations involves finding all angle values within a specified interval that satisfy the given equation. This often requires using identities, algebraic manipulation, and understanding the periodic nature of trigonometric functions to find all valid solutions.
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Sine Function Properties and Identities

The sine function is periodic with period 2π and has symmetry properties such as sin(α) = sin(π - α). Recognizing these properties helps in solving equations like sin x = sin 2x by expressing one angle in terms of the other or using identities to simplify the equation.
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Graph of Sine and Cosine Function

Interval Notation and Solution Sets

When solving trigonometric equations, solutions must be restricted to the given interval, such as [0, 2π) for radians or [0°, 360°) for degrees. Understanding how to express solutions within these intervals ensures the answers are complete and conform to the problem's requirements.
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i & j Notation
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