In Exercises 12–18, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places.sin 2x + cos x = 0

Verified step by step guidance

1. Start by using the double angle identity for sine: \( \sin 2x = 2 \sin x \cos x \). Substitute this into the equation to get: \( 2 \sin x \cos x + \cos x = 0 \).
2. Factor out \( \cos x \) from the equation: \( \cos x (2 \sin x + 1) = 0 \).
3. Set each factor equal to zero and solve for \( x \): \( \cos x = 0 \) and \( 2 \sin x + 1 = 0 \).
4. Solve \( \cos x = 0 \) for \( x \) in the interval \([0, 2\pi)\). The solutions are \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).
5. Solve \( 2 \sin x + 1 = 0 \) for \( x \). This simplifies to \( \sin x = -\frac{1}{2} \). Find the values of \( x \) in the interval \([0, 2\pi)\) where \( \sin x = -\frac{1}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, angle sum and difference identities, and double angle formulas. In this problem, recognizing that sin(2x) can be expressed using the double angle identity sin(2x) = 2sin(x)cos(x) is crucial for simplifying the equation.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy the equation within a specified interval. This often requires isolating the trigonometric function and using inverse functions or identities to find solutions. In this case, we need to manipulate the equation to express it in terms of a single trigonometric function, allowing us to find the values of x that satisfy the equation within the interval [0, 2π).

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. The interval [0, 2π) indicates that the solutions must be greater than or equal to 0 and less than 2π. Understanding this notation is essential for determining the valid solutions to the equation, as it restricts the possible values of x to a specific range, which is important for trigonometric functions that are periodic.