Solve for the exact solutions in the interval $[0, 2\pi)$. List your answers separated by a comma, if it has no real solutions, enter DNE.

$2\sqrt{3} \sin \left( \frac{x}{2} \right) = 3$

Neetesh Kumar | October 15, 2024

Pre-Calculus Homework Help

Assignment 9.5 Question 18: - on Solving Trignometric Equations
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Step-by-step solution:

Step 1: Solve for $\sin \left( \frac{x}{2} \right)$

First, divide both sides of the equation by $2\sqrt{3}$:

$\sin \left( \frac{x}{2} \right) = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2}$

Step 2: Solve for $\frac{x}{2}$

The general solution for $\sin(\theta) = \frac{\sqrt{3}}{2}$ occurs at:

$\frac{x}{2} = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad \frac{x}{2} = \pi - \frac{\pi}{3} + 2n\pi$

This simplifies to:

$\frac{x}{2} = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad \frac{x}{2} = \frac{2\pi}{3} + 2n\pi$

Step 3: Multiply both sides by 2 to solve for $x$

Multiply both solutions by 2:

$x = \frac{2\pi}{3} + 4n\pi \quad \text{or} \quad x = \frac{4\pi}{3} + 4n\pi$

Step 4: Find solutions in the interval $[0, 2\pi)$

We now calculate the values of $x$ for $n = 0$:

For the first solution:

$x = \frac{2\pi}{3}$

For the second solution:

$x = \frac{4\pi}{3}$

Thus, the solutions are:

$x = \frac{2\pi}{3}, \frac{4\pi}{3}$

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Final Answer:

$\boxed{\frac{2\pi}{3}, \frac{4\pi}{3}}$

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