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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 124
Capitolo 3, Problema 124

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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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.
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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).
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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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