Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
sin x = sin 2x

검증된 단계별 안내

Start with the given equation: \(\sin x = \sin 2x\).

Recall the identity that if \(\sin A = \sin B\), then the solutions are given by two cases: \(A = B + 2k\pi\) or \(A = \pi - B + 2k\pi\), where \(k\) is any integer.

Apply this identity to the equation by setting \(x = 2x + 2k\pi\) and \(x = \pi - 2x + 2k\pi\) separately.

Solve each resulting equation for \(x\) within the interval \([0, 2\pi)\) by isolating \(x\) and considering integer values of \(k\) that keep \(x\) in the interval.

List all values of \(x\) found from both cases that lie within \([0, 2\pi)\) as the exact solutions.

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.

Trigonometric Equation Solving

Solving trigonometric equations involves finding all angle values within a specified interval that satisfy the given equation. This often requires using identities, algebraic manipulation, and understanding the periodic nature of trigonometric functions to find all valid solutions.

Sine Function Properties and Identities

The sine function is periodic with period 2π and has symmetry properties such as sin(α) = sin(π - α). Recognizing these properties helps in solving equations like sin x = sin 2x by expressing one angle in terms of the other or using identities to simplify the equation.

Interval Notation and Solution Sets

When solving trigonometric equations, solutions must be restricted to the given interval, such as [0, 2π) for radians or [0°, 360°) for degrees. Understanding how to express solutions within these intervals ensures the answers are complete and conform to the problem's requirements.

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.sin x = sin 2x