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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 67
Capitolo 3, Problema 67

In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). 4 cos² x = 5 - 4 sin x

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Start by recalling the Pythagorean identity: \(\cos^{2} x = 1 - \sin^{2} x\). This allows us to rewrite the equation in terms of \(\sin x\) only.
Substitute \(\cos^{2} x\) with \(1 - \sin^{2} x\) in the equation: \(4(1 - \sin^{2} x) = 5 - 4 \sin x\).
Expand the left side: \(4 - 4 \sin^{2} x = 5 - 4 \sin x\).
Bring all terms to one side to form a quadratic equation in \(\sin x\): \(-4 \sin^{2} x + 4 \sin x + 4 - 5 = 0\), which simplifies to \(-4 \sin^{2} x + 4 \sin x - 1 = 0\).
Multiply the entire equation by \(-1\) to make the quadratic coefficient positive: \(4 \sin^{2} x - 4 \sin x + 1 = 0\). Now, solve this quadratic equation for \(\sin x\) using the quadratic formula.

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Pythagorean Identity

The Pythagorean identity states that sin²x + cos²x = 1 for any angle x. This identity allows you to express cos²x in terms of sin²x or vice versa, which is essential for rewriting the given equation into a single trigonometric function to simplify solving.
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Solving Trigonometric Equations

Solving trigonometric equations involves manipulating the equation using identities and algebraic techniques to isolate the trigonometric function. Then, find all angle solutions within the specified interval by considering the unit circle and periodicity of sine and cosine.
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How to Solve Linear Trigonometric Equations

Interval Restriction and General Solutions

When solving trigonometric equations, solutions must be found within a given interval, here [0, 2π). This requires understanding the unit circle and how sine and cosine values repeat every 2π, ensuring all valid solutions in the interval are identified and extraneous ones discarded.
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