Question

The inverse sine, inverse cosine, and inverse tangent functions have the following domains and ranges. (a) The function $\sin ^{-1}$ has domain ___ and range _. (b) The function $\cos ^{-1}$ has domain _ and range _. (c) The function $\tan ^{-1}$ has domain _ and range ___.

Name: Problem 2, Chapter 6: Precalculus Mathematics for Calculus
Uploaded: 2020-05-19T20:22:36-08:00
Duration: 1 min 23 s
Channel: Ramzi Deek
Description: The inverse sine, inverse cosine, and inverse tangent functions have the following domains and ranges. (a) The function sin^-1 has domain and range . (b) The function cos^-1 has domain and range . (c) The function tan^-1 has domain and range .

   The inverse sine, inverse cosine, and inverse tangent functions have the following domains and ranges.
(a) The function $\sin ^{-1}$ has domain _____ and range _____.
(b) The function $\cos ^{-1}$ has domain _____ and range _____.
(c) The function $\tan ^{-1}$ has domain _____ and range _____.

Precalculus Mathematics for Calculus

James Stewart,… 7th Edition

Chapter 6, Problem 2

Instant Answer

Solved by Expert Ramzi Deek

05/19/2020

Step 1

The domain of a function is the set of all possible input values (x-values), which produce a valid output from a particular function. The range of a function is the set of possible output values (y-values), which result from the function. For the inverse sine Show more…

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The inverse sine, inverse cosine, and inverse tangent functions have the following domains and ranges. (a) The function $\sin ^{-1}$ has domain _____ and range _____. (b) The function $\cos ^{-1}$ has domain _____ and range _____. (c) The function $\tan ^{-1}$ has domain _____ and range _____.

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Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions return the angle associated with a given trigonometric value. They are defined by restricting the domain of the original trigonometric functions so that they become one-to-one, which is necessary for the existence of an inverse function.

Domain and Range in Inverse Functions

The domain of an inverse trigonometric function is determined by the range of its corresponding trigonometric function, while its range is a specifically chosen interval (the principal branch) that ensures the function is single-valued. This restriction is essential to avoid ambiguity in the output values.

Principal Branches

The principal branch of an inverse trigonometric function is the selected interval where the original trigonometric function is bijective. This selection guarantees that every output, that is every angle, is unique, making it possible to systematically solve equations involving these functions.

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