Textbook Question
Find the exact value of each real number y if it exists. Do not use a calculator.
y = arctan 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 19
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 19Chapter 7, Problem 19
Solve each equation for exact solutions over the interval [0, 2π).
tan² x + 3 = 0
Verified step by step guidance1
Start with the given equation: \(\tan^{2} x + 3 = 0\).
Isolate the \(\tan^{2} x\) term by subtracting 3 from both sides: \(\tan^{2} x = -3\).
Recall that \(\tan^{2} x\) represents the square of the tangent function, which is always greater than or equal to zero for all real \(x\) because squaring any real number cannot produce a negative result.
Since \(\tan^{2} x = -3\) implies a negative value, recognize that there are no real values of \(x\) for which this equation holds true within the interval \([0, 2\pi)\).
Conclude that the equation has no exact solutions in the interval \([0, 2\pi)\) because the square of tangent cannot be negative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of the Tangent Function
The tangent function, tan(x), is defined as sin(x)/cos(x) and has a period of π. It is undefined where cos(x) = 0, and its values can be positive or negative. Understanding its behavior and domain is essential when solving equations involving tan²(x).
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Introduction to Tangent Graph
Solving Quadratic Equations in Trigonometric Functions
Equations like tan²(x) + 3 = 0 can be treated as quadratic in terms of tan(x). Solving involves isolating tan²(x) and checking for real solutions, considering that squares of real numbers are non-negative, which affects the existence of solutions.
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6:24Solving Quadratic Equations by Completing the Square
Interval and Exact Solutions in Trigonometry
Finding exact solutions over [0, 2π) requires identifying all angles within one full rotation that satisfy the equation. Solutions must be expressed in exact form (like π/4), and understanding the periodicity and symmetry of trig functions helps determine all valid solutions.
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4:34How to Solve Linear Trigonometric Equations
Related Practice
Textbook Question
Solve each equation for x, where x is restricted to the given interval.
y = 1/2 cot 3 x , for x in [0, π/3]
Textbook Question
Use a calculator to approximate each value in decimal degrees.
θ = cos⁻¹ 0.80396577
Textbook Question
Use a calculator to approximate each value in decimal degrees.
θ = arctan 1.7804675
Textbook Question
Find the exact value of each real number y if it exists. Do not use a calculator.
y = tan⁻¹ 1
Textbook Question
Find the exact value of each real number y if it exists. Do not use a calculator.
y = arcsin (―√3/2)