Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problema 6.3.12

Capitolo 7, Problema 6.3.12

Solve for exact solutions over the interval [0°, 360°).
sin θ/2 = -√3/2

Guida verificata passo dopo passo

Start by rewriting the equation: \(\frac{\sin \theta}{2} = -\frac{\sqrt{3}}{2}\). Multiply both sides by 2 to isolate \(\sin \theta\): \(\sin \theta = -\sqrt{3}\).

Recognize that the sine function has a range of \([-1, 1]\), so \(\sin \theta = -\sqrt{3}\) is not possible because \(\sqrt{3} > 1\). This suggests there might be a misunderstanding in the original equation.

Re-examine the problem statement: if the equation is \(\sin \frac{\theta}{2} = -\frac{\sqrt{3}}{2}\), then focus on solving for \(\frac{\theta}{2}\) first.

Recall that \(\sin x = -\frac{\sqrt{3}}{2}\) corresponds to angles where the sine value is negative and equal to \(-\frac{\sqrt{3}}{2}\). The reference angle for \(\sin x = \frac{\sqrt{3}}{2}\) is \(60^\circ\), so the solutions for \(\sin x = -\frac{\sqrt{3}}{2}\) are in the third and fourth quadrants: \(x = 180^\circ + 60^\circ = 240^\circ\) and \(x = 360^\circ - 60^\circ = 300^\circ\).

Set \(\frac{\theta}{2} = 240^\circ\) and \(\frac{\theta}{2} = 300^\circ\), then multiply both sides by 2 to find \(\theta\): \(\theta = 480^\circ\) and \(\theta = 600^\circ\). Since the interval is \([0^\circ, 360^\circ)\), subtract \(360^\circ\) to find equivalent angles within the interval: \(\theta = 120^\circ\) and \(\theta = 240^\circ\).

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Solving Basic Trigonometric Equations

To solve equations like sin(θ/2) = -√3/2, identify the angles where the sine function equals the given value. This involves recalling the unit circle values and understanding that sine is negative in specific quadrants. Solutions are found by setting the argument (θ/2) equal to these reference angles.

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Unit Circle and Reference Angles

The unit circle helps determine exact sine values at standard angles. Reference angles are acute angles related to the given angle, used to find sine values in different quadrants. Since sine is negative in the third and fourth quadrants, these guide where solutions lie for sin(θ/2) = -√3/2.

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Reference Angles on the Unit Circle

Interval Considerations and Back-Substitution

The problem restricts θ to [0°, 360°), so after solving for θ/2, multiply solutions by 2 to find θ. Ensure the final θ values fall within the given interval by considering the domain and periodicity of sine. This step is crucial to list all valid solutions without extraneous ones.

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Solve Trig Equations Using Identity Substitutions

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