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Inverse Trigonometric Functions Recall some facts about inverse functions: 1.For a function to have an inverse it must be a one-to-one function. 2.The domain of f -1 is the range of f. 3.The range of f -1 is the domain of f. 4.If ( a,b ) is a point on the graph of f, then ( b,a ) is a point on the graph of f -1. 5. f(f -1 ) = x for all x in the domain of f -1. 6. f -1 (f ) = x for all x in the domain of f.

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The graph of y = sin x is given below. Since y = sin x is not a 1-1 function on its domain of all real numbers, it does not have an inverse. Note that it fails the horizontal line test.

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We must restrict the domain of the sine function so that it will be a 1-1 function, and have an inverse. This could be done in several ways, but the following is most common:

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Definition: Inverse Sine

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Another way of writing the inverse of the sine function is as follows:

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The graph of y = cos x is given below. Since y = cos x is not a 1-1 function on its domain of all real numbers, it does not have an inverse. Note that it fails the horizontal line test.

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We must restrict the domain of the cosine function so that it will be a 1-1 function, and have an inverse. This could be done in several ways, but the following is most common:

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Definition: Inverse Cosine

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Another way of writing the inverse of the cosine function is as follows:

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The graph of y = tan x is given below. Since y = tan x is not a 1-1 function on its normal domain, it does not have an inverse. Note that it fails the horizontal line test.

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We must restrict the domain of the tangent function so that it will be a 1-1 function, and have an inverse. This could be done in several ways, but the following is most common:

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Definition: Inverse Tangent

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Another way of writing the inverse of the tangent function is as follows:

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Slide 5.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Slide 5.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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