객관식
Find all solutions to the equation.
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11. Inverse Trigonometric Functions and Basic Trig Equations
Linear Trigonometric Equations
객관식
Find all solutions to the equation.
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검증된 단계별 안내1
Identify the given equation: \( \sin \theta = -\frac{\sqrt{3}}{2} \). This indicates that we are looking for angles where the sine value is \(-\frac{\sqrt{3}}{2}\).
Recall that the sine function is negative in the third and fourth quadrants. The reference angle for \( \sin \theta = \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{3} \).
Determine the angles in the third and fourth quadrants that have the same reference angle. These are \( \theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \) and \( \theta = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \).
Express the general solutions for these angles by adding multiples of the full period of the sine function, which is \( 2\pi \). Thus, the solutions are \( \theta = \frac{4\pi}{3} + 2\pi n \) and \( \theta = \frac{5\pi}{3} + 2\pi n \), where \( n \) is an integer.
Verify the solutions by checking that they satisfy the original equation \( \sin \theta = -\frac{\sqrt{3}}{2} \) and are consistent with the sine function's periodicity and quadrant rules.
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