Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.
sin² θ + 3 sin θ + 2 = 0

검증된 단계별 안내

Recognize that the equation is a quadratic in terms of \(\sin \theta\). Rewrite the equation as \(\sin^{2} \theta + 3 \sin \theta + 2 = 0\) to emphasize this.

Let \(x = \sin \theta\). Substitute into the equation to get a quadratic equation: \(x^{2} + 3x + 2 = 0\).

Factor the quadratic equation: find two numbers that multiply to 2 and add to 3. This gives \((x + 1)(x + 2) = 0\).

Solve each factor for \(x\): \(x + 1 = 0\) gives \(x = -1\), and \(x + 2 = 0\) gives \(x = -2\). Since \(\sin \theta\) must be between -1 and 1, discard \(x = -2\).

Find all angles \(\theta\) in the interval \([0^\circ, 360^\circ)\) such that \(\sin \theta = -1\). Use the unit circle or inverse sine function to determine these values.

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주요 개념

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

Solving Quadratic Equations in Trigonometric Form

The given equation sin²θ + 3 sinθ + 2 = 0 is a quadratic equation in terms of sinθ. To solve it, treat sinθ as a variable (e.g., x), solve the quadratic equation for x, and then find θ values that satisfy sinθ = x within the given interval.

Unit Circle and Sine Values

Understanding the unit circle is essential to find all angles θ where sinθ equals a specific value. Since sine corresponds to the y-coordinate on the unit circle, solutions for sinθ = x can be found by identifying angles in [0°, 360°) with that sine value.

Interval Restriction and Solution Representation

The problem restricts θ to the interval [0°, 360°), so only solutions within one full rotation are valid. Solutions should be expressed as exact values (like 30°, 45°) or rounded to the nearest tenth of a degree, depending on whether the sine values correspond to standard angles.

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.sin² θ + 3 sin θ + 2 = 0