In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin 2x cos x + cos 2x sin x = √ 2/2

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1. Recognize that the left side of the equation, \(\sin 2x \cos x + \cos 2x \sin x\), matches the form of the sine addition formula: \(\sin A \cos B + \cos A \sin B = \sin(A + B)\). Here, let \(A = 2x\) and \(B = x\).
2. Apply the sine addition formula to rewrite the left side as a single sine function: \(\sin(2x + x) = \sin 3x\).
3. Rewrite the equation as \(\sin 3x = \frac{\sqrt{2}}{2}\).
4. Recall that \(\sin \theta = \frac{\sqrt{2}}{2}\) at specific angles within \([0, 2\pi)\), namely \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{3\pi}{4}\), plus any full rotations of \(2\pi\).
5. Set \(3x = \frac{\pi}{4} + 2k\pi\) and \(3x = \frac{3\pi}{4} + 2k\pi\) for integers \(k\), then solve for \(x\) by dividing both sides by 3. Finally, determine all solutions for \(x\) within the interval \([0, 2\pi)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They allow simplification or transformation of expressions, such as the product-to-sum or angle addition formulas, which are essential for solving complex trigonometric equations.

Angle Addition Formulas

Angle addition formulas express the sine or cosine of a sum of angles in terms of sines and cosines of individual angles. For example, sin(A + B) = sin A cos B + cos A sin B. Recognizing these patterns helps rewrite and simplify equations to find solutions.

Solving Trigonometric Equations on a Given Interval

Solving trigonometric equations on a specific interval, such as [0, 2π), involves finding all angle values within that range that satisfy the equation. This requires understanding periodicity, principal values, and sometimes using inverse trigonometric functions to determine all valid solutions.