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

검증된 단계별 안내

Recognize that the equation is \( \cos 2x = -1 \), where the argument of the cosine function is \( 2x \). Our goal is to find all \( x \) in the interval \( [0, 2\pi) \) that satisfy this equation.

Recall the general solution for \( \cos \theta = -1 \). Cosine equals \( -1 \) at \( \theta = \pi + 2k\pi \), where \( k \) is any integer.

Set \( 2x = \pi + 2k\pi \) to match the argument of the cosine function in the equation. This gives the equation \( 2x = \pi + 2k\pi \).

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

Find all values of \( x \) within the interval \( [0, 2\pi) \) by substituting integer values of \( k \) and checking which \( x \) values fall into the interval.

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주요 개념

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Double-Angle Identity for Cosine

The double-angle identity states that cos(2x) can be expressed in terms of x as cos(2x) = 2cos²(x) - 1 or cos(2x) = 1 - 2sin²(x). This identity allows us to rewrite or interpret the equation involving cos(2x) in terms of a single angle x, facilitating the solving process.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values within a given interval that satisfy the equation. For cos(2x) = -1, we determine the angles where the cosine function equals -1, then solve for x by considering the periodicity and domain restrictions.

Interval and Domain Considerations

When solving trigonometric equations over a specific interval, such as [0, 2π), it is essential to find all solutions within that range. Since the equation involves 2x, the interval for 2x becomes [0, 4π), and solutions must be adjusted accordingly to find all valid x values in the original interval.

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