In Exercises 53–62, solve each equation on the interval [0, 2𝝅). sin x + 2 sin x cos x = 0

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Start with the given equation: \(\sin x + 2 \sin x \cos x = 0\).

Factor out the common factor \(\sin x\) from both terms: \(\sin x (1 + 2 \cos x) = 0\).

Set each factor equal to zero to find possible solutions: \(\sin x = 0\) and \(1 + 2 \cos x = 0\).

Solve \(\sin x = 0\) on the interval \([0, 2\pi)\), which occurs where \(x = 0, \pi, 2\pi\) (considering the interval endpoint).

Solve \(1 + 2 \cos x = 0\) by isolating \(\cos x\): \(\cos x = -\frac{1}{2}\), then find all \(x\) in \([0, 2\pi)\) where this is true.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Equations

Trigonometric equations involve expressions with trigonometric functions like sine and cosine. Solving these equations means finding all angle values within a given interval that satisfy the equation. Understanding how to manipulate and simplify these equations is essential for finding correct solutions.

Trigonometric Identities

Trigonometric identities are equations true for all values of the variable where both sides are defined. Identities like the product-to-sum or double-angle formulas help simplify complex expressions. For example, recognizing that 2 sin x cos x equals sin 2x can transform the equation into a simpler form.

Solving Equations on a Specific Interval

When solving trigonometric equations, solutions must be found within a specified interval, here [0, 2π). This requires understanding the periodic nature of trig functions and identifying all valid solutions within one full cycle of the unit circle.

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