Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.
3 cos² θ + 2 cos θ - 1 = 0

검증된 단계별 안내

Recognize that the given equation is a quadratic in terms of \( \cos \theta \): \( 3 \cos^{2} \theta + 2 \cos \theta - 1 = 0 \). To solve it, let \( x = \cos \theta \), so the equation becomes \( 3x^{2} + 2x - 1 = 0 \).

Use the quadratic formula to solve for \( x \): \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \), where \( a = 3 \), \( b = 2 \), and \( c = -1 \).

Calculate the discriminant \( \Delta = b^{2} - 4ac \) and then find the two possible values for \( x = \cos \theta \).

For each value of \( x \), determine the corresponding angles \( \theta \) in the interval \( [0^{\circ}, 360^{\circ}) \) by using the inverse cosine function: \( \theta = \cos^{-1}(x) \). Remember that cosine is positive in Quadrants I and IV, and negative in Quadrants II and III, so find all angles that satisfy each \( x \).

Write the solutions as exact values if possible, or round to the nearest tenth of a degree as appropriate, ensuring all solutions lie within the given interval.

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주요 개념

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

Quadratic Equations in Trigonometric Functions

This concept involves treating trigonometric equations like algebraic quadratics by substituting the trigonometric function (e.g., cos θ) as a variable. Solving the quadratic yields possible values for the function, which can then be used to find the angle θ.

Unit Circle and Angle Measurement

Understanding the unit circle helps relate cosine values to specific angles within the interval [0°, 360°). Each cosine value corresponds to one or two angles in this range, depending on the sign and magnitude, which is essential for finding all solutions.

Exact Values and Approximation in Trigonometry

Some trigonometric solutions correspond to well-known exact values (like cos 60° = 0.5), while others require approximation using a calculator. Recognizing when to use exact values or round to the nearest tenth degree ensures precise and appropriate answers.

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.3 cos² θ + 2 cos θ - 1 = 0