Explanation Given: Evaluate csc − 1 ( csc − π ) . Restrictions on the Domains of the Trig Functions: A function must be one-to-one for it to have an inverse. As we know, the trigonometric functions are not one-to-one and in fact they are periodic. So in order to define inverse functions we need to restrict the domain of each trigonometric function to a region in which it is one-to-one but also attains all of its values. We do this by selecting a specific period for each function and using this as a domain on which an inverse can be defined. Standard Restricted Domains Function Domain Range sin ( x ) [ − π 2 , π 2 ] [ − 1 , 1 ] cos ( x ) [ 0 , π ] [ − 1 , 1 ] tan ( x ) ( − π 2 , π 2 ) ( − ∞ , ∞ ) cot ( x ) ( 0 , π ) ( − ∞ , ∞ ) sec ( x ) [ 0 , π 2 ) ∪ ( π 2 , π ] ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) csc ( x ) [ − π 2 , 0 ) ∪ ( 0 , π 2 ] ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) Definitions of the Inverse Trigonometric Functions When the trigonometric functions are restricted to the above domains, they become one-to-one functions. For example the inverse of sine function is denoted by sin − 1 ( x ) (or) arcsin ( x ) . Both are read as “arc sine”. Standard Restricted Domains Function Domain Range sin − 1 ( x ) [ − 1 , 1 ] [ − π 2 , π 2 ] cos − 1 ( x ) [ − 1 , 1 ] [ 0 , π ] tan − 1 ( x ) ( − ∞ , ∞ ) ( − π 2 , π 2 ) cot − 1 ( x ) ( − ∞ , ∞ ) ( 0 , π ) sec − 1 ( x ) ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) [ 0 , π 2 ) ∪ ( π 2 , π ] csc − 1 ( x ) ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) [ − π 2 , 0 ) ∪ ( 0 , π 2 ] Properties of Inverse Trigonometric Functions: Function sin − 1 ( sin x ) = x x ∈ [ − π 2 , π 2 ] cos − 1 ( cos x ) = x x ∈

3rd Edition

ISBN: 9781464114885

Author: Jon Rogawski, Colin Adams

Publisher: W. H. Freeman

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Question

Chapter 1.5, Problem 41E

To determine

The value of $\csc^{- 1} (\csc - π)$ without using a calculator

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Chapter 1.5, Problem 40E

Chapter 1.5, Problem 42E

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Chapter 1 Solutions

Calculus: Early Transcendentals

Ch. 1.1 - Prob. 1PQ Ch. 1.1 - Prob. 2PQ Ch. 1.1 - Prob. 3PQ Ch. 1.1 - Prob. 4PQ Ch. 1.1 - Prob. 5PQ Ch. 1.1 - Prob. 6PQ Ch. 1.1 - Prob. 7PQ Ch. 1.1 - Prob. 8PQ Ch. 1.1 - Prob. 9PQ Ch. 1.1 - Prob. 10PQ

Ch. 1.1 - Prob. 1E Ch. 1.1 - Prob. 2E Ch. 1.1 - Prob. 3E Ch. 1.1 - Prob. 4E Ch. 1.1 - Prob. 5E Ch. 1.1 - Prob. 6E Ch. 1.1 - Prob. 7E Ch. 1.1 - Prob. 8E Ch. 1.1 - Prob. 9E Ch. 1.1 - Prob. 10E Ch. 1.1 - Prob. 11E Ch. 1.1 - Prob. 12E Ch. 1.1 - Prob. 13E Ch. 1.1 - Prob. 14E Ch. 1.1 - Prob. 15E Ch. 1.1 - Prob. 16E Ch. 1.1 - Prob. 17E Ch. 1.1 - Prob. 18E Ch. 1.1 - Prob. 19E Ch. 1.1 - Prob. 20E Ch. 1.1 - Prob. 21E Ch. 1.1 - Prob. 22E Ch. 1.1 - Prob. 23E Ch. 1.1 - Prob. 24E Ch. 1.1 - Prob. 25E Ch. 1.1 - Prob. 26E Ch. 1.1 - Prob. 27E Ch. 1.1 - Prob. 28E Ch. 1.1 - Prob. 29E Ch. 1.1 - Prob. 30E Ch. 1.1 - Prob. 31E Ch. 1.1 - Prob. 32E Ch. 1.1 - Prob. 33E Ch. 1.1 - Prob. 34E Ch. 1.1 - Prob. 35E Ch. 1.1 - Prob. 36E Ch. 1.1 - Prob. 37E Ch. 1.1 - Prob. 38E Ch. 1.1 - Prob. 39E Ch. 1.1 - Prob. 40E Ch. 1.1 - Prob. 41E Ch. 1.1 - Prob. 42E Ch. 1.1 - Prob. 43E Ch. 1.1 - Prob. 44E Ch. 1.1 - Prob. 45E Ch. 1.1 - Prob. 46E Ch. 1.1 - Prob. 47E Ch. 1.1 - Prob. 48E Ch. 1.1 - Prob. 49E Ch. 1.1 - Prob. 50E Ch. 1.1 - 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The value of csc − 1 ( csc − π ) without using a calculator

The value of csc−1(csc−π) without using a calculator

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Chapter 1 Solutions

The value of $\csc^{- 1} (\csc - π)$ without using a calculator