MathLabs

Problem 1

Determine all values of xx in the interval 0≤x≤2pi0 \le x \le 2\\pi which satisfy 2cosx≤∣sqrt1+sin2x−sqrt1−sin2x∣≤sqrt22\\cos x \le |\\sqrt{1+\\sin 2x}-\\sqrt{1-\\sin 2x}| \le \\sqrt{2}.
Step 2 of 4: Resolve the two sign cases
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cos2xge0Longrightarrowy=2∣sinx∣.quadcos2x<0Longrightarrowy=2∣cosx∣.\\cos 2x\\ge0\\Longrightarrow y=2|\\sin x|.\\quad \\cos 2x<0\\Longrightarrow y=2|\\cos x|.
Detailed analysis

When cos2xge0\\cos 2x\\ge0, use 2−2cos2x=4sin2x2-2\\cos 2x=4\\sin^2x. When cos2x<0\\cos 2x<0, use 2+2cos2x=4cos2x2+2\\cos 2x=4\\cos^2x; the nonnegative square root gives the displayed formulas.