Textbook Question
Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate
tan 2x + sec 2x = 3
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 6.2.41
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 6.2.41Chapter 7, Problem 6.2.41
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
Verified step by step guidance1
Start by recognizing that the given equation is a quadratic in terms of \(\tan \theta\): \(\tan^{2} \theta + 4 \tan \theta + 2 = 0\).
Use the quadratic formula to solve for \(\tan \theta\). Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=1\), \(b=4\), and \(c=2\) in this case.
Calculate the discriminant \(\Delta = b^{2} - 4ac = 4^{2} - 4 \times 1 \times 2\) and then find the two possible values for \(\tan \theta\) using the quadratic formula.
Once you have the two values for \(\tan \theta\), find the corresponding angles \(\theta\) by taking the arctangent (inverse tangent) of each value. Remember to consider the interval \([0^\circ, 360^\circ)\) and the periodicity and sign of the tangent function to find all solutions within this interval.
Write the solutions for \(\theta\) either as exact values (if they correspond to special angles) or rounded to the nearest tenth of a degree, as appropriate.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Quadratic Equations
The given equation is quadratic in terms of tan θ, so solving it involves using methods like factoring, completing the square, or the quadratic formula to find values of tan θ that satisfy the equation.
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6:24
Solving Quadratic Equations by Completing the Square
Properties of the Tangent Function
Understanding the tangent function's behavior, including its period of 180°, domain restrictions, and how to interpret solutions for θ within the interval [0°, 360°), is essential for finding all valid angle solutions.
Recommended video:
5:43Introduction to Tangent Graph
Inverse Tangent and Angle Solutions
After finding values for tan θ, use the inverse tangent function to determine θ. Since tangent is periodic, consider all angles in [0°, 360°) that correspond to the solutions, adjusting for the function's periodicity.
Recommended video:
3:17Inverse Tangent
Related Practice
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.
3 csc² x/2 = 2 sec x
Textbook Question
Solve for exact solutions over the interval [0°, 360°).
cos θ/2 = -1/2
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
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