Solve the equation for solutions in the interval $[0,2 \pi)$. Use algebraic methods and give exact values. Support your solutions graphically.
\[
\sin \left(\frac{x}{2}\right)=1-\sin \left(\frac{x}{2}\right)
\]
Answer
Final Answer: The solution to the equation is \(\boxed{x \approx 2.094}\).
Step 1 ：We are given the equation \(\sin \left(\frac{x}{2}\right)=1-\sin \left(\frac{x}{2}\right)\).
Step 2 ：We can simplify this equation to \(2\sin \left(\frac{x}{2}\right)=1\).
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 \([0,2 \pi)\).
Step 4 ：The solution to the equation \(\sin \left(\frac{x}{2}\right)=1-\sin \left(\frac{x}{2}\right)\) in the interval \([0,2 \pi)\) is \(x \approx 2.094\).
Step 5 ：Final Answer: The solution to the equation is \(\boxed{x \approx 2.094}\).
