Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

2 - sin 2θ = 4 sin 2θ

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Start by rewriting the given equation: \(2 - \sin 2\theta = 4 \sin 2\theta\).

Combine like terms by adding \(\sin 2\theta\) to both sides to isolate the sine term: \(2 = 4 \sin 2\theta + \sin 2\theta\) which simplifies to \(2 = 5 \sin 2\theta\).

Solve for \(\sin 2\theta\) by dividing both sides by 5: \(\sin 2\theta = \frac{2}{5}\).

Find the general solutions for \(2\theta\) by using the inverse sine function: \(2\theta = \sin^{-1}\left(\frac{2}{5}\right)\) and also consider the second solution in the range \([0, 2\pi)\), which is \(2\theta = \pi - \sin^{-1}\left(\frac{2}{5}\right)\).

Divide all solutions for \(2\theta\) by 2 to solve for \(\theta\), then express the answers in radians and degrees. For approximate answers, round radians to four decimal places and degrees to the nearest tenth. Also, write the solutions using the least possible nonnegative angle measures.

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