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Algebra Difficulty 4.6 AIME Find the answer

Find all values of xx with 0≤x<2π0 \leq x<2 \pi that satisfy sin⁡x+cos⁡x=2\sin x+\cos x=\sqrt{2}.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Squaring both sides gives sin⁡2x+cos⁡2x+2sin⁡xcos⁡x=1+sin⁡2x=2\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x=1+\sin 2 x=2, so x=π4,5π4x=\frac{\pi}{4}, \frac{5 \pi}{4}.

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