HMMT February

Algebra Difficulty 4.6 AIME Find the answer

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

Your answer

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

Solution

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

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