In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. __ sin 2x = √ 3 sin x

Verified step by step guidance

1. Step 1: Use the double angle identity for sine, which is $\sin 2x=2\sin x\cos x$. Substitute this into the equation to get $2\sin x\cos x=\sqrt{3}\sin x$.
2. Step 2: Factor out $\sin x$ from both sides of the equation: $\sin x(2\cos x-\sqrt{3})=0$.
3. Step 3: Set each factor equal to zero and solve for $x$: $\sin x=0$ and $2\cos x-\sqrt{3}=0$.
4. Step 4: Solve $\sin x=0$ for $x$ in the interval $[0,2\pi )$. The solutions are $x=0,\pi ,2\pi$.
5. Step 5: Solve $2\cos x=\sqrt{3}$ for $x$ in the interval $[0,2\pi )$. This simplifies to $\cos x=\frac{\sqrt{3}}{2}$, which gives solutions $x=\frac{\pi }{6},\frac{11\pi }{6}$.

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 identities. In this problem, recognizing that sin(2x) can be expressed as 2sin(x)cos(x) is crucial for simplifying the equation.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy the given 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 find values of x that satisfy sin(2x) = √3 sin(x) 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 within the range starting from 0 (inclusive) to 2π (exclusive). Understanding this notation is essential for determining the valid solutions for x in the context of the problem.