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

Verified step by step guidance

1. Start with the given equation: \(\sin \frac{x}{2} - \cos \frac{x}{2} = 0\).
2. Rearrange the equation to isolate one trigonometric function: \(\sin \frac{x}{2} = \cos \frac{x}{2}\).
3. Divide both sides by \(\cos \frac{x}{2}\) (assuming \(\cos \frac{x}{2} \neq 0\)) to get \(\tan \frac{x}{2} = 1\).
4. 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 \(x \in [0, 2\pi)\).
5. Multiply the solutions for \(\frac{x}{2}\) by 2 to find the corresponding values of \(x\) in the interval \([0, 2\pi)\).

Key Concepts

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.