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Sagot :
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]
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]
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