Textbook Question
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
tan² θ + 4 tan θ + 2 = 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 6.2.63
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 6.2.63Chapter 7, Problem 6.2.63
The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval [0, 2π). Express solutions to four decimal places.
x² + sin x - x³ - cos x = 0
Verified step by step guidance1
Rewrite the given equation as a function to analyze: define \( f(x) = x^{2} + \sin x - x^{3} - \cos x \).
Use a graphing calculator or graphing software to plot the function \( f(x) \) over the interval \( [0, 2\pi) \).
Identify the points where the graph of \( f(x) \) crosses the x-axis within the interval \( [0, 2\pi) \). These x-values are the solutions to the equation.
Use the calculator's root-finding feature (such as the zero or root function) to find the x-values of these intersections, ensuring to specify the interval around each root for accuracy.
Record each solution to four decimal places as required, making sure all solutions within \( [0, 2\pi) \) are included.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Transcendental Equations
Transcendental equations involve both algebraic and transcendental functions like sine and cosine, making them unsolvable by standard algebraic methods. These equations often require numerical or graphical techniques to find approximate solutions.
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Parameterizing Equations
Graphing and Root-Finding Techniques
Graphing calculators plot functions to visually identify where they cross the x-axis, indicating solutions. Root-finding methods such as the bisection or Newton-Raphson method can then be used to approximate these solutions to a desired precision.
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Interval and Solution Precision
Specifying the interval [0, 2π) restricts the domain to one full cycle of trigonometric functions, ensuring all relevant solutions are found within this range. Expressing solutions to four decimal places requires careful numerical approximation and rounding.
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Related Practice
Textbook Question
Which one of the following equations has solution π?
a. arccos (―1) = x
b. arccos 1 = x
c. arcsin (―1) = x
Textbook Question
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
sec² θ tan θ = 2 tan θ
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.
8 sec² x/2 = 4
Textbook Question
Solve each equation for all exact solutions, in degrees.
2√3 cos (θ/2) = -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.
√2 cos 2θ = -1