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Unit 4 Section 5

# Unit 4 Section 5 - 4.5 INVERSE CIRCULAR FUNCTIONS From unit...

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4.5 INVERSE CIRCULAR FUNCTIONS From unit 2, we learned that any relation has an inverse (relation), implying that all functions have inverses. We also learned that for an inverse of a function to be in itself a function, the function must me one-to-one or injective. Recall : A function is 1-1 if and only if for any and in the domain of , if , then ; or if the graph of satisfies the conditions the horizontal line test. TIME TO THINK! Are circular functions one-to-one? Prove or disprove using the definition of 1-1 functions. Graphically, it is obvious that all circular functions are not one-to-one. Now, let us look at the inverses of each circular function Definition 4.5.1 Let be the sine function defined by . Then the inverse sine relation or arcsine , denoted by arcsin , is defined by where Example 4.5.1 Since arcsine is not a function, an arcsin x has several “values”, that is TIME TO THINK! Give a generalization for the “values” of the following: In spite of the use of the word “ arc ”, arcsin is to be interpreted as stating that “ is an arclength whose sine is ”. Similarly, we will be using this prefix for the inverse relations of the other circular functions.

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Figure 1. The Graph of the Inverse Sine Relation Definition 4.5.2 Let be the sine function defined by . Then the inverse sine relation or arccossine , denoted by arccos , is defined by where TIME TO THINK!
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Unit 4 Section 5 - 4.5 INVERSE CIRCULAR FUNCTIONS From unit...

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