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

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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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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.

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Introduction to Quadratic Equations

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.

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Introduction to the Unit Circle

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.

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