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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 1
Capitolo 7, Problema 1

Which one of the following equations has solution 0?
a. arctan 1 = x
b. arccos 0 = x
c. arcsin 0 = x

Guida verificata passo dopo passo
1
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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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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Evaluate Composite Functions - Values Not on Unit Circle

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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Introduction to Inverse Trig Functions
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