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.

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

Neetesh Kumar | October 15, 2024

Pre-Calculus Homework Help

Assignment 9.5 Question 19: - on Solving Trignometric Equations
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Step-by-step solution:

Step 1: Combine like terms

Add $\sin \left( \frac{x}{2} \right)$ to both sides to combine the sine terms:

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

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

Divide both sides by 2:

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

Step 3: Find general solutions for $\sin \left( \frac{x}{2} \right) = \frac{\sqrt{2}}{2}$

The general solutions for $\sin(\theta) = \frac{\sqrt{2}}{2}$ occur at:

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

This simplifies to:

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

Step 4: Multiply both sides by 2 to solve for $x$

Multiply both solutions by 2:

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

Step 5: Find solutions in the interval $[0, 2\pi]$

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

For the first solution:

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

For the second solution:

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

Thus, the solutions are:

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

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Final Answer:

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

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