In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin x cos x = √ 2 / 4

Verified step by step guidance

1. Start with the given equation: \(\sin x \cos x = \frac{\sqrt{2}}{4}\).
2. Recall the double-angle identity for sine: \(\sin(2x) = 2 \sin x \cos x\). Use this to rewrite the left side of the equation.
3. Multiply both sides of the equation by 2 to express it in terms of \(\sin(2x)\): \(2 \sin x \cos x = 2 \times \frac{\sqrt{2}}{4}\), which simplifies to \(\sin(2x) = \frac{\sqrt{2}}{2}\).
4. Solve the equation \(\sin(2x) = \frac{\sqrt{2}}{2}\) for \$2x\( on the interval \([0, 4\pi)\), since \)x\( is in \([0, 2\pi)\) and the argument is \)2x$.
5. Find all values of \(x\) by dividing the solutions for \$2x$ by 2, ensuring the solutions fall within the original interval \([0, 2\pi)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, using the double-angle identity for sine, sin(2x) = 2 sin x cos x, simplifies the equation and helps solve for x efficiently.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a specified interval. After applying identities, one must consider the periodic nature of sine and cosine to find all valid solutions in [0, 2π).

Interval and General Solutions

When solving trig equations on a specific interval like [0, 2π), it is important to find all solutions within that range. Since trig functions are periodic, multiple angles can satisfy the equation, so understanding how to restrict solutions to the given interval is essential.