Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.

sin x/2 - cos x/2 = 0

검증된 단계별 안내

Start with the given equation: \(\sin \frac{x}{2} - \cos \frac{x}{2} = 0\).

Rearrange the equation to isolate one trigonometric function: \(\sin \frac{x}{2} = \cos \frac{x}{2}\).

Divide both sides by \(\cos \frac{x}{2}\) (assuming \(\cos \frac{x}{2} \neq 0\)) to get \(\tan \frac{x}{2} = 1\).

Solve for \(\frac{x}{2}\) by finding the angles where \(\tan \theta = 1\) within the interval for \(\frac{x}{2}\), which is \([0, \pi)\) because \(x \in [0, 2\pi)\).

Multiply the solutions for \(\frac{x}{2}\) by 2 to find the corresponding values of \(x\) in the interval \([0, 2\pi)\).

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주요 개념

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

Trigonometric Equations

Trigonometric equations involve functions like sine and cosine and require finding all angle values that satisfy the equation within a given interval. Solving these often involves algebraic manipulation and applying identities to isolate the variable.

Angle Division and Substitution

When the variable appears as a fraction of the angle (e.g., x/2), it is helpful to use substitution (such as letting t = x/2) to simplify the equation. This allows solving for t first, then converting back to x, ensuring solutions fit the original interval.

Using Trigonometric Identities

Identities like sin A = cos A or sin A = cos(π/2 - A) help transform and simplify equations. Recognizing that sin θ = cos θ implies θ = π/4 + kπ enables finding exact solutions efficiently.

Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.sin x/2 - cos x/2 = 0