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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 6.3.33
Capitolo 7, Problema 6.3.33

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
cos 2x + cos x = 0

Guida verificata passo dopo passo
1
Start with the given equation: \(\cos 2x + \cos x = 0\).
Use the double-angle identity for cosine: \(\cos 2x = 2\cos^2 x - 1\). Substitute this into the equation to get \(2\cos^2 x - 1 + \cos x = 0\).
Rewrite the equation as a quadratic in terms of \(\cos x\): \(2\cos^2 x + \cos x - 1 = 0\).
Solve the quadratic equation \(2y^2 + y - 1 = 0\) where \(y = \cos x\). Use the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a=2\), \(b=1\), and \(c=-1\).
Find the values of \(x\) in the interval \([0, 2\pi)\) (or \(\theta\) in \([0^\circ, 360^\circ)\)) by solving \(\cos x = y\) for each root \(y\) obtained, using inverse cosine and considering all solutions within the given interval.

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Double-Angle Identity for Cosine

The double-angle identity expresses cos(2x) in terms of cos(x) and sin(x), commonly as cos(2x) = 2cos²(x) - 1. This identity helps rewrite the equation cos(2x) + cos(x) = 0 into a single trigonometric function, simplifying the solving process.
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Double Angle Identities

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angles within the given interval that satisfy the equation. This often requires using identities, factoring, or substitution to find exact solutions.
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How to Solve Linear Trigonometric Equations

Interval and Angle Measurement

Understanding the specified interval [0, 2π) for x in radians and [0°, 360°) for θ in degrees is crucial. Solutions must be found within these ranges, ensuring all valid angles are included and expressed exactly, not approximately.
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Reference Angles on the Unit Circle
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Solve each equation for all exact solutions, in degrees.

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Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

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