In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x + sin x = 0

Verified step by step guidance

1. Start by recognizing the given equation: \(\sin 2x + \sin x = 0\). Our goal is to solve for \(x\) in the interval \([0, 2\pi)\).
2. Use the double-angle identity for sine: \(\sin 2x = 2 \sin x \cos x\). Substitute this into the equation to get \(2 \sin x \cos x + \sin x = 0\).
3. Factor out the common factor \(\sin x\): \(\sin x (2 \cos x + 1) = 0\). This product equals zero, so set each factor equal to zero separately.
4. Solve the first equation \(\sin x = 0\) for \(x\) in \([0, 2\pi)\), and then solve the second equation \(2 \cos x + 1 = 0\) for \(x\) in the same interval.
5. For \(2 \cos x + 1 = 0\), isolate \(\cos x\) to get \(\cos x = -\frac{1}{2}\). Find all \(x\) in \([0, 2\pi)\) where \(\cos x = -\frac{1}{2}\). Combine these solutions with those from \(\sin x = 0\) to get the complete solution set.

Key Concepts

Double-Angle Identity for Sine

The double-angle identity expresses sin(2x) as 2 sin(x) cos(x). This allows rewriting the equation sin 2x + sin x = 0 into a form involving sin(x) and cos(x), facilitating factorization and solution. Recognizing and applying this identity simplifies trigonometric equations.

Solving Trigonometric Equations Using Factoring

Factoring trigonometric expressions helps break down complex equations into simpler parts. After rewriting sin 2x + sin x = 0 using identities, factoring out common terms like sin(x) can isolate factors, each set to zero, yielding possible solutions within the given interval.

Finding Solutions on a Specific Interval

Trigonometric equations often have multiple solutions within a given interval, such as [0, 2π). It is essential to find all solutions that satisfy the equation in this range, using exact values when possible or approximations rounded to four decimal places, ensuring completeness and accuracy.