Textbook Question
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.
sin 2θ = cos 2θ +1
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.RE.37
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 6.RE.37Chapter 7, Problem 6.RE.37
Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.
2 tan x -1 = 0
Verified step by step guidance1
Start with the given equation: \(2 \tan x - 1 = 0\).
Isolate \(\tan x\) by adding 1 to both sides and then dividing by 2, giving \(\tan x = \frac{1}{2}\).
Recall that \(\tan x = \frac{\sin x}{\cos x}\) and that the tangent function has a period of \(\pi\), so solutions repeat every \(\pi\) radians.
Find the principal solution \(x_1\) by taking the arctangent: \(x_1 = \arctan\left(\frac{1}{2}\right)\), which will be in the first quadrant since \(\frac{1}{2}\) is positive.
Find the second solution \(x_2\) in the interval \([0, 2\pi)\) by adding \(\pi\) to the principal solution: \(x_2 = x_1 + \pi\). These two values are the solutions to the equation in the given interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Trigonometric Equations
Solving trigonometric equations involves isolating the trigonometric function and finding all angle values within a given interval that satisfy the equation. This often requires algebraic manipulation and understanding the periodic nature of trig functions.
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Properties of the Tangent Function
The tangent function, tan(x), is periodic with period π and is undefined at odd multiples of π/2. Knowing its period and behavior helps find all solutions within the interval by adding integer multiples of π to the principal solution.
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Interval Notation and Solution Restrictions
When solving trig equations over a specific interval like [0, 2π), solutions must be restricted to that range. This means identifying all valid solutions within the interval and expressing them as exact values or decimal approximations as required.
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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 of a degree, as appropriate.
sin² θ + 3 sin θ + 2 = 0
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√3 sin x/2 = 3
Textbook Question
Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.
tan x = cot x
Textbook Question
Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.
tan² 2x -1 = 0
Textbook Question
Solve each equation for all exact solutions, in radians.
cos 2x + cos x = 0