What are all values of e, 0leqslant Theta leqslant 2pi for which (sin^2Theta )(2cosTheta -sqrt (2))geqslant 0 Select all that apply.

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What are all values of e, 0leqslant Theta leqslant 2pi for which (sin^2Theta )(2cosTheta -sqrt (2))geqslant 0 Select all that apply.

Answer 1

The values of \( \theta \) that satisfy the inequality are in the intervals [0, π/4] and [7π/4, 2π], and the individual solutions are 0, π/4, and 2π.

Answer 2

## Step1 The given inequality is: \[ \left(\sin ^{2} \theta\right)(2 \cos \theta-\sqrt{2}) \geq 0 \] To find the solution set, we need to find the values of \( \theta \) for which the expression is equal to zero. ## Step2 The expression will be zero when either \( \sin \theta = 0 \) or \( 2\cos \theta - \sqrt{2} = 0 \). ## Step3 For \( \sin \theta = 0 \), the solutions are \( \theta = 0, \pi, 2\pi \). ## Step4 For \( 2\cos \theta - \sqrt{2} = 0 \), the solution is \( \theta = \frac{\pi}{4}, \frac{7\pi}{4} \). ## Step5 These solutions split the interval [0, 2π] into five intervals: [0, π/4], (π/4, π), (π, 7π/4), (7π/4, 2π). ## Step6 We can test each interval to see whether the inequality holds. ## Step7 For \( \theta \) in [0, π/4], the expression is always ≥ 0. ## Step8 For \( \theta \) in (π/4, π), the expression is always ≤ 0. ## Step9 For \( \theta \) in (π, 7π/4), the expression is always ≤ 0. ## Step10 For \( \theta \) in (7π/4, 2π), the expression is always ≥ 0. ## Step11 Also, the solutions are \( \theta = 0, \frac{\pi}{4}, 2\pi \).

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What are all values of e, 0leqslant Theta leqslant 2pi for which (sin^2Theta )(2cosTheta -sqrt (2))geqslant 0 Select all that apply.

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