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

검증된 단계별 안내

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 \(B = x\).

Apply the sine addition formula to rewrite the left side as a single sine function: \(\sin(2x + x) = \sin 3x\).

Rewrite the equation as \(\sin 3x = \frac{\sqrt{2}}{2}\).

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 \(2\pi\).

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 \([0, 2\pi)\).

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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.