To solve the equation \(\sin(2x) = \sqrt{2} \cos(x)\) on the interval \([0,2\pi)\), we'll start by using trigonometric identities to simplify and solve the equation step-by-step.
1. Expand \(\sin(2x)\) using the double-angle formula:
[tex]\[
\sin(2x) = 2\sin(x)\cos(x)
\][/tex]
Thus, the given equation becomes:
[tex]\[
2 \sin(x) \cos(x) = \sqrt{2} \cos(x)
\][/tex]
2. Rearrange the equation and factor:
[tex]\[
2 \sin(x) \cos(x) - \sqrt{2} \cos(x) = 0
\][/tex]
Factor out \(\cos(x)\):
[tex]\[
\cos(x) (2 \sin(x) - \sqrt{2}) = 0
\][/tex]
3. Solve for \(\cos(x) = 0\):
[tex]\[
\cos(x) = 0
\][/tex]
The values of \(x\) that satisfy \(\cos(x) = 0\) are:
[tex]\[
x = \frac{\pi}{2}, \frac{3\pi}{2}
\][/tex]
4. Solve for \(2 \sin(x) - \sqrt{2} = 0\):
[tex]\[
2 \sin(x) = \sqrt{2}
\][/tex]
[tex]\[
\sin(x) = \frac{\sqrt{2}}{2}
\][/tex]
The values of \(x\) that satisfy \(\sin(x) = \frac{\sqrt{2}}{2}\) are:
[tex]\[
x = \frac{\pi}{4}, \frac{3\pi}{4}
\][/tex]
5. List all solutions within the given interval \([0, 2\pi)\):
We combine the solutions from both parts:
[tex]\[
x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{4}, \frac{3\pi}{4}
\][/tex]
6. Verify the solutions lie in the desired interval and order them:
[tex]\[
x = \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}
\][/tex]
Thus, the solutions to the equation \(\sin(2x) = \sqrt{2} \cos(x)\) on the interval \([0, 2\pi)\) are:
[tex]\[
x = \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}
\][/tex]
Additionally, we can express the numerical values of these solutions:
[tex]\[
x \approx 0.785, 1.571, 2.356, 4.712
\][/tex]
Only those within \([0, 2\pi)\) are:
[tex]\[
x \approx 0.785398163397448, 1.57079632679490, 2.35619449019234
\][/tex]
So, confirming the valid solutions in the interval are:
[tex]\[
x \approx \boxed{0.785398163397448, 1.57079632679490, 2.35619449019234}
\][/tex]
