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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 6.3.25
Capitolo 7, Problema 6.3.25

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/2) = √2 ― sin (x/2)

Guida verificata passo dopo passo
1
Start by rewriting the given equation: \(\sin\left(\frac{x}{2}\right) = \sqrt{2} - \sin\left(\frac{x}{2}\right)\).
Add \(\sin\left(\frac{x}{2}\right)\) to both sides to combine like terms: \(\sin\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right) = \sqrt{2}\), which simplifies to \(2 \sin\left(\frac{x}{2}\right) = \sqrt{2}\).
Divide both sides by 2 to isolate the sine term: \(\sin\left(\frac{x}{2}\right) = \frac{\sqrt{2}}{2}\).
Recall that \(\sin(\alpha) = \frac{\sqrt{2}}{2}\) at angles \(\alpha = \frac{\pi}{4}\) and \(\alpha = \frac{3\pi}{4}\) within the interval \([0, 2\pi)\), and similarly for degrees \(45^\circ\) and \(135^\circ\) within \([0^\circ, 360^\circ)\).
Set \(\frac{x}{2} = \frac{\pi}{4}\) and \(\frac{x}{2} = \frac{3\pi}{4}\) (or \(\frac{x}{2} = 45^\circ\) and \(\frac{x}{2} = 135^\circ\)), then solve for \(x\) (or \(\theta\)) by multiplying both sides by 2. Make sure to check that the solutions lie within the original intervals \([0, 2\pi)\) for \(x\) and \([0^\circ, 360^\circ)\) for \(\theta\).

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Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle values within a specified interval that satisfy the equation. This often requires algebraic manipulation and using known values or identities to simplify the equation.
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Properties of the Sine Function

The sine function is periodic with period 2π and has a range of [-1, 1]. Understanding its symmetry and key values at standard angles helps in finding exact solutions. For example, sin(θ) = sin(π - θ) and sin(θ) = sin(θ + 2πk) for any integer k.
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Graph of Sine and Cosine Function

Interval Notation and Angle Measurement

The problem specifies solutions over intervals [0, 2π) for radians and [0°, 360°) for degrees. Knowing how to convert between radians and degrees and restricting solutions to these intervals ensures all valid solutions are found without repetition.
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i & j Notation
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