Textbook Question
Which one of the following equations has solution 3π/4
a. arctan 1 = x
b. arcsin √2/2 = x
c. arccos (―√2 /2) = x
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 1
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 1Chapter 7, Problem 1
Which one of the following equations has solution 0?
a. arctan 1 = x
b. arccos 0 = x
c. arcsin 0 = x
Verified step by step guidance1
Understand that the problem asks which equation has the solution \( x = 0 \). This means we need to find which inverse trigonometric function equals zero for the given input.
Recall the definitions of the inverse trigonometric functions: \( \arctan y = x \) means \( \tan x = y \), \( \arccos y = x \) means \( \cos x = y \), and \( \arcsin y = x \) means \( \sin x = y \).
For option (a), \( \arctan 1 = x \) means \( \tan x = 1 \). Determine if \( x = 0 \) satisfies \( \tan 0 = 1 \).
For option (b), \( \arccos 0 = x \) means \( \cos x = 0 \). Check if \( x = 0 \) satisfies \( \cos 0 = 0 \).
For option (c), \( \arcsin 0 = x \) means \( \sin x = 0 \). Verify if \( x = 0 \) satisfies \( \sin 0 = 0 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, such as arcsin, arccos, and arctan, return the angle whose trigonometric value is given. For example, arcsin(x) gives the angle whose sine is x. Understanding their definitions and ranges is essential to determine the angle solutions.
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Introduction to Inverse Trig Functions
Evaluating Specific Values of Inverse Trigonometric Functions
Certain values of inverse trig functions correspond to well-known angles. For instance, arcsin(0) = 0 because sin(0) = 0, arccos(0) = π/2 because cos(π/2) = 0, and arctan(1) = π/4 because tan(π/4) = 1. Recognizing these helps identify which equation equals zero.
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Range and Principal Values of Inverse Trigonometric Functions
Each inverse trig function has a principal range where it returns unique values: arcsin ranges from -π/2 to π/2, arccos from 0 to π, and arctan from -π/2 to π/2. Knowing these ranges is crucial to correctly interpret the solutions and determine if zero is a valid output.
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Related Practice
Textbook Question
Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).
<IMAGE>
cos x = √3/2
Textbook Question
Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).
<IMAGE>
cos x = 1/2
Textbook Question
Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).
<IMAGE>
sin x = ―1/2