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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 114
Capitolo 3, Problema 114

In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 7 cos x = 4 - 2 sin² x

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Rewrite the given equation: \(7 \cos x = 4 - 2 \sin^{2} x\).
Use the Pythagorean identity \(\sin^{2} x = 1 - \cos^{2} x\) to express everything in terms of \(\cos x\). Substitute to get: \(7 \cos x = 4 - 2(1 - \cos^{2} x)\).
Simplify the right side: \(7 \cos x = 4 - 2 + 2 \cos^{2} x\), which becomes \(7 \cos x = 2 + 2 \cos^{2} x\).
Rearrange the equation to standard quadratic form in terms of \(\cos x\): \(2 \cos^{2} x - 7 \cos x + 2 = 0\).
Solve the quadratic equation for \(\cos x\) using the quadratic formula \(\cos x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) where \(a=2\), \(b=-7\), and \(c=2\). Then find all \(x\) in \([0, 2\pi)\) such that \(\cos x\) equals the solutions found.

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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 sin²x + cos²x = 1 is essential to rewrite sin²x in terms of cos²x, enabling the equation to be expressed in a single trigonometric function for easier solving.
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Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a specified interval. This often requires algebraic manipulation, use of identities, and understanding the periodic nature of sine and cosine to find all valid solutions between 0 and 2π.
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Interval and Exact vs Approximate Solutions

The problem restricts solutions to the interval [0, 2π), meaning only angles within one full rotation are considered. Solutions should be given as exact values (like π/3) when possible, or approximated to four decimal places when exact forms are complicated or unavailable, ensuring clarity and precision.
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