Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
2 tan θ sin θ - tan θ = 0

검증된 단계별 안내

Start by writing down the given equation: \(2 \tan \theta \sin \theta - \tan \theta = 0\).

Factor out the common term \(\tan \theta\) from the left side: \(\tan \theta (2 \sin \theta - 1) = 0\).

Set each factor equal to zero to find possible solutions: \(\tan \theta = 0\) and \(2 \sin \theta - 1 = 0\).

Solve \(\tan \theta = 0\) by finding all angles \(\theta\) in \([0^\circ, 360^\circ)\) where the tangent function is zero.

Solve \(2 \sin \theta - 1 = 0\) by isolating \(\sin \theta\) to get \(\sin \theta = \frac{1}{2}\), then find all angles \(\theta\) in \([0^\circ, 360^\circ)\) that satisfy this.

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주요 개념

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

Trigonometric Equations

Trigonometric equations involve functions like sine, cosine, and tangent. Solving these requires isolating the trigonometric function and finding all angle values within a given interval that satisfy the equation.

Factoring and Zero-Product Property

Factoring expressions allows breaking down complex equations into simpler products. The zero-product property states that if a product equals zero, at least one factor must be zero, enabling separate equations to be solved.

Solving for Angles in a Given Interval

When solving trigonometric equations over [0°, 360°), it is essential to find all angle solutions within one full rotation. This involves using reference angles and considering the signs of trig functions in each quadrant.

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.2 tan θ sin θ - tan θ = 0