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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 6.3.31
Capitolo 7, Problema 6.3.31

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

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Start by rewriting the given equation: \(\sin\left(\frac{\theta}{2}\right) = \csc\left(\frac{\theta}{2}\right)\). Recall that \(\csc x = \frac{1}{\sin x}\), so substitute to get \(\sin\left(\frac{\theta}{2}\right) = \frac{1}{\sin\left(\frac{\theta}{2}\right)}\).
Multiply both sides of the equation by \(\sin\left(\frac{\theta}{2}\right)\) to eliminate the fraction, giving \(\sin^2\left(\frac{\theta}{2}\right) = 1\).
Recognize that \(\sin^2 x = 1\) implies \(\sin x = \pm 1\). So, set \(\sin\left(\frac{\theta}{2}\right) = 1\) and \(\sin\left(\frac{\theta}{2}\right) = -1\) separately to find all possible solutions.
Solve each equation for \(\frac{\theta}{2}\) over the interval corresponding to \(\theta \in [0^\circ, 360^\circ)\), which means \(\frac{\theta}{2} \in [0^\circ, 180^\circ)\). Use the unit circle values where sine equals \(1\) or \(-1\) within this interval.
Finally, multiply each solution for \(\frac{\theta}{2}\) by 2 to find the values of \(\theta\) in the interval \([0^\circ, 360^\circ)\). These will be the exact solutions to the original equation.

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Trigonometric Functions and Their Definitions

Understanding sine (sin) and cosecant (csc) functions is essential. Sine of an angle is the ratio of the opposite side to the hypotenuse in a right triangle, while cosecant is its reciprocal, defined as 1/sin. Recognizing this reciprocal relationship helps in transforming and solving the equation.
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Solving Trigonometric Equations

Solving equations like sin(θ/2) = csc(θ/2) involves rewriting the equation using identities, simplifying, and finding all angle solutions within the given interval. This process often requires considering domain restrictions and checking for extraneous solutions.
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Interval and Angle Measurement Conventions

The problem specifies solutions over [0, 2π) radians and [0°, 360°) degrees. Understanding how to convert between radians and degrees and how to interpret these intervals ensures that all valid solutions are found and expressed correctly within the specified domain.
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