Solve for exact solutions over the interval [0, 2π).

cos 2x = 1/2

검증된 단계별 안내

Recognize that the equation is \( \cos 2x = \frac{1}{2} \). Our goal is to find all values of \( x \) in the interval \( [0, 2\pi) \) that satisfy this equation.

Recall that \( \cos \theta = \frac{1}{2} \) at specific standard angles. Specifically, \( \cos \theta = \frac{1}{2} \) when \( \theta = \frac{\pi}{3} \) and \( \theta = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \).

Set \( 2x = \theta \), so the solutions for \( 2x \) are \( \frac{\pi}{3} + 2k\pi \) and \( \frac{5\pi}{3} + 2k\pi \), where \( k \) is any integer, because cosine is periodic with period \( 2\pi \).

Solve for \( x \) by dividing each solution by 2: \( x = \frac{\pi}{6} + k\pi \) and \( x = \frac{5\pi}{6} + k\pi \).

Find all values of \( x \) within the interval \( [0, 2\pi) \) by substituting integer values of \( k \) (such as 0 and 1) into the expressions and checking which values lie in the interval.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 relate the equation cos(2x) = 1/2 to standard cosine values and solve for x.

Solving Trigonometric Equations

Solving equations like cos(2x) = 1/2 involves finding all angles within the given interval where the cosine value matches 1/2. This requires understanding the unit circle, reference angles, and periodicity of cosine to determine all valid solutions.

Interval and Domain Considerations

Since the problem restricts solutions to the interval [0, 2π), it is essential to adjust for the double angle by considering the domain of 2x and then mapping solutions back to x within the original interval. This ensures all exact solutions are correctly identified.

Solve for exact solutions over the interval [0, 2π).cos 2x = 1/2