In Exercises 121–126, solve each equation on the interval [0, 2𝝅). 3 cos² x - sin x = cos² x

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Start by rewriting the given equation: \(3 \cos^{2} x - \sin x = \cos^{2} x\).

Bring all terms to one side to set the equation equal to zero: \(3 \cos^{2} x - \sin x - \cos^{2} x = 0\), which simplifies to \(2 \cos^{2} x - \sin x = 0\).

Use the Pythagorean identity \(\cos^{2} x = 1 - \sin^{2} x\) to express everything in terms of \(\sin x\): substitute to get \(2(1 - \sin^{2} x) - \sin x = 0\).

Expand and simplify the equation: \(2 - 2 \sin^{2} x - \sin x = 0\), then rearrange to form a quadratic in \(\sin x\): \(-2 \sin^{2} x - \sin x + 2 = 0\).

Multiply the entire equation by \(-1\) to make the quadratic standard: \(2 \sin^{2} x + \sin x - 2 = 0\). Now solve this quadratic equation for \(\sin x\) within the interval \([0, 2\pi)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, the Pythagorean identity, such as cos²x + sin²x = 1, is essential to rewrite and simplify expressions involving cos²x and sin x.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a specified interval. This requires algebraic manipulation and understanding how to find angles that satisfy the equation on [0, 2π).

Interval and General Solutions in Trigonometry

When solving trigonometric equations, solutions are often found over a specific interval, such as [0, 2π). Understanding how to determine all valid solutions within this interval, including using reference angles and symmetry properties of sine and cosine, is crucial.

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Fundamental Trigonometric Identities

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