Solve the equation for solutions in the interval $[0, 2 π)$ . Use algebraic methods and give exact values. Support your solutions graphically.
$\sin (\frac{x}{2}) = 1 - \sin (\frac{x}{2})$

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Answer

Final Answer: The solution to the equation is $x \approx 2.094$ .

Steps

Step 1 ：We are given the equation $\sin (\frac{x}{2}) = 1 - \sin (\frac{x}{2})$ .

Step 2 ：We can simplify this equation to $2 \sin (\frac{x}{2}) = 1$ .

Step 3 ：We solve this equation for $\frac{x}{2}$ and then multiply the result by 2 to find the solutions for x in the interval $[0, 2 π)$ .

Step 4 ：The solution to the equation $\sin (\frac{x}{2}) = 1 - \sin (\frac{x}{2})$ in the interval $[0, 2 π)$ is $x \approx 2.094$ .

Step 5 ：Final Answer: The solution to the equation is $x \approx 2.094$ .