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

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Solving Trigonometric Equations

This involves isolating the trigonometric function and finding all angle values that satisfy the equation within a given domain. Solutions often include general forms using periodicity, and exact or approximate values depending on the problem's requirements.

Double-Angle Identities

Double-angle identities express trigonometric functions of 2θ in terms of θ, such as sin(2θ) = 2 sin θ cos θ. Recognizing and using these identities helps simplify equations and find solutions more efficiently.

Angle Measurement and Conversion

Understanding the difference between radians and degrees, and converting between them, is essential. Solutions must be expressed in the correct units, rounded appropriately, and within the specified interval, often using the least nonnegative angle measure.