**Understanding Domains for Inverse Trigonometric Functions**The image contains three graphs illustrating the functions \( f(x) = \sin(x) \), \( f(x) = \cos(x) \), and \( f(x) = \tan(x) \). Each graph represents the standard trigonometric function necessary to form their respective inverse functions. Let's examine each graph in detail:1. **Graph of \( f(x) = \sin(x) \)**: - The sine function graph oscillates between -1 and 1 along the y-axis. - The period of the sine function is \( 2\pi \), meaning it repeats every \( 2\pi \) units. - The key segment that is typically highlighted to create the inverse sine function, \( \sin^{-1}(x) \) or arcsin(x), is usually from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).2. **Graph of \( f(x) = \cos(x) \)**: - The cosine function also oscillates between -1 and 1. - Like the sine function, it has a period of \( 2\pi \). - For the inverse cosine function, \( \cos^{-1}(x) \) or arccos(x), the domain typically used is from 0 to \(\pi\).3. **Graph of \( f(x) = \tan(x) \)**: - The tangent function displays vertical asymptotes where the function is undefined, occurring at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \). - The pattern repeats every \( \pi \) units. - For the inverse tangent function, \( \tan^{-1}(x) \) or arctan(x), the highlighted domain is usually from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).Each graph supports the visual understanding of the trigonometric function's behavior over its important domain, which is used for deriving the inverse function. These graphs are essential for understanding how the domains are restricted to ensure that the inverse trigonometric functions are bijective and thus properly defined.

Answered: Image /qna-images/answer/5cd375f9-ade6-487d-96af-f9724e677c6f.jpg

Highlight each graph below depicting the domain used to create each inverse trigonometric function. x Clear 211 Undo f(x) = sin(x) 10 5 -5 -10 Redo 211 -21 f(x) = cos(x) 10 -5 -5 -10 211 201 f(x) =tan(x) 10 -5 10 211

Highlight each graph below depicting the domain used to create each inverse trigonometric function. x Clear 211 Undo f(x) = sin(x) 10 5 -5 -10 Redo 211 -21 f(x) = cos(x) 10 -5 -5 -10 211 201 f(x) =tan(x) 10 -5 10 211

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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Question

**Understanding Domains for Inverse Trigonometric Functions**

The image contains three graphs illustrating the functions \( f(x) = \sin(x) \), \( f(x) = \cos(x) \), and \( f(x) = \tan(x) \). Each graph represents the standard trigonometric function necessary to form their respective inverse functions. Let's examine each graph in detail:

1. **Graph of \( f(x) = \sin(x) \)**:
- The sine function graph oscillates between -1 and 1 along the y-axis.
- The period of the sine function is \( 2\pi \), meaning it repeats every \( 2\pi \) units.
- The key segment that is typically highlighted to create the inverse sine function, \( \sin^{-1}(x) \) or arcsin(x), is usually from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

2. **Graph of \( f(x) = \cos(x) \)**:
- The cosine function also oscillates between -1 and 1.
- Like the sine function, it has a period of \( 2\pi \).
- For the inverse cosine function, \( \cos^{-1}(x) \) or arccos(x), the domain typically used is from 0 to \(\pi\).

3. **Graph of \( f(x) = \tan(x) \)**:
- The tangent function displays vertical asymptotes where the function is undefined, occurring at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \).
- The pattern repeats every \( \pi \) units.
- For the inverse tangent function, \( \tan^{-1}(x) \) or arctan(x), the highlighted domain is usually from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Each graph supports the visual understanding of the trigonometric function's behavior over its important domain, which is used for deriving the inverse function.

These graphs are essential for understanding how the domains are restricted to ensure that the inverse trigonometric functions are bijective and thus properly defined.

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