Textbook Question
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
Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
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Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 6.2.35
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 6.2.35Chapter 7, Problem 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
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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Related Practice
Textbook Question
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
csc² θ ―2 cot θ = 0
Textbook Question
Solve each equation for exact solutions.
tan⁻¹ x - tan⁻¹ (1/x ) = π/6
Textbook Question
Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.
2 - sin 2θ = 4 sin 2θ
Textbook Question
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
cot θ + 2 csc θ = 3
Textbook Question
Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
3 tan 3x = √3