Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). sin 2x = √3 / 2

Verified step by step guidance

1. Start by writing down the given equation: \(\sin 2x = \frac{\sqrt{3}}{2}\).
2. Recall that \(\sin \theta = \frac{\sqrt{3}}{2}\) at specific standard angles. Identify all angles \(\theta\) in the interval \([0, 2\pi)\) where this is true. These angles are \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{2\pi}{3}\).
3. Since the equation involves \$2x\(, set \(2x = \frac{\pi}{3} + 2k\pi\) and \(2x = \frac{2\pi}{3} + 2k\pi\), where \)k$ is any integer, to account for the periodicity of the sine function.
4. Solve each equation for \(x\) by dividing both sides by 2: \(x = \frac{\pi}{6} + k\pi\) and \(x = \frac{\pi}{3} + k\pi\).
5. Find all values of \(x\) within the interval \([0, 2\pi)\) by substituting integer values of \(k\) (such as \(k=0\) and \(k=1\)) into the expressions for \(x\).

Key Concepts

Double-Angle Identities

Double-angle identities express trigonometric functions of multiples of an angle in terms of single angles. For example, sin(2x) = 2 sin x cos x. These identities help simplify and solve equations involving multiple angles like sin 2x.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a given interval. This often requires using inverse functions and considering the periodicity of sine, cosine, or tangent.

Unit Circle and Interval Constraints

The unit circle helps visualize sine and cosine values for angles between 0 and 2π. Understanding the interval [0, 2π) is crucial to find all valid solutions within one full rotation, considering the periodic nature of trigonometric functions.