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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 6.2.35
Capitolo 7, Problema 6.2.35

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
2 tan θ sin θ - tan θ = 0

Guida verificata passo dopo passo
1
Start by writing down the given equation: \(2 \tan \theta \sin \theta - \tan \theta = 0\).
Factor out the common term \(\tan \theta\) from the left side: \(\tan \theta (2 \sin \theta - 1) = 0\).
Set each factor equal to zero to find possible solutions: \(\tan \theta = 0\) and \(2 \sin \theta - 1 = 0\).
Solve \(\tan \theta = 0\) by finding all angles \(\theta\) in \([0^\circ, 360^\circ)\) where the tangent function is zero.
Solve \(2 \sin \theta - 1 = 0\) by isolating \(\sin \theta\) to get \(\sin \theta = \frac{1}{2}\), then find all angles \(\theta\) in \([0^\circ, 360^\circ)\) that satisfy this.

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Trigonometric Equations

Trigonometric equations involve functions like sine, cosine, and tangent. Solving these requires isolating the trigonometric function and finding all angle values within a given interval that satisfy the equation.
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How to Solve Linear Trigonometric Equations

Factoring and Zero-Product Property

Factoring expressions allows breaking down complex equations into simpler products. The zero-product property states that if a product equals zero, at least one factor must be zero, enabling separate equations to be solved.
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Solving for Angles in a Given Interval

When solving trigonometric equations over [0°, 360°), it is essential to find all angle solutions within one full rotation. This involves using reference angles and considering the signs of trig functions in each quadrant.
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Solving Problems with Complementary & Supplementary Angles
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