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

2 cos 2x = √3

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Start by isolating the trigonometric function in the equation: given \( 2 \cos 2x = \sqrt{3} \), divide both sides by 2 to get \( \cos 2x = \frac{\sqrt{3}}{2} \).

Recall the general solutions for \( \cos \theta = \frac{\sqrt{3}}{2} \). The cosine function equals \( \frac{\sqrt{3}}{2} \) at angles \( \theta = \frac{\pi}{6} \) and \( \theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} \) within the interval \( [0, 2\pi) \).

Since the argument of the cosine is \( 2x \), set \( 2x = \frac{\pi}{6} + 2k\pi \) and \( 2x = \frac{11\pi}{6} + 2k\pi \), where \( k \) is any integer, to account for the periodicity of cosine.

Solve each equation for \( x \) by dividing both sides by 2: \( x = \frac{\pi}{12} + k\pi \) and \( x = \frac{11\pi}{12} + k\pi \).

Find all values of \( x \) within the interval \( [0, 2\pi) \) by substituting integer values of \( k \) (such as 0 and 1) and discarding any solutions outside the interval.

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주요 개념

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Double-Angle Identity for Cosine

The double-angle identity expresses cos(2x) in terms of x, commonly as cos(2x) = 2cos²(x) - 1 or cos(2x) = cos²(x) - sin²(x). Recognizing this helps rewrite or interpret the equation involving cos(2x) and solve for x within the given interval.

Solving Trigonometric Equations

Solving equations like 2 cos(2x) = √3 involves isolating the trigonometric function, finding the general solutions using inverse cosine, and then determining all solutions within the specified interval by considering the periodicity of cosine.

Interval and Angle Measurement Conventions

Understanding the domain restrictions, such as x in [0, 2π) radians or θ in [0°, 360°), is essential to find all valid solutions within one full rotation. This ensures solutions are expressed exactly and appropriately within the given interval.

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.2 cos 2x = √3