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Pre-Calc Exam Notes 126

# Pre-Calc Exam Notes 126 - − π 2 ≤ y ≤ π 2 y n= 0...

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126 Chapter 5 Graphing and Inverse Functions §5.3 The inverse functions for cotangent, cosecant, and secant can be determined by looking at their graphs. For example, the function y = cot x is one-to-one in the interval (0 , π ), where it has a range equal to the set of all real numbers. Thus, the inverse cotangent y = cot 1 x is a function whose domain is the set of all real numbers and whose range is the interval (0 , π ). In other words: cot 1 (cot y ) = y for 0 < y < π (5.8) cot (cot 1 x ) = x for all real x (5.9) The graph of y = cot 1 x is shown below in Figure 5.3.11. x y 0 π 2 π π 4 π 2 3 π 4 3 π 4 π 4 π 2 y = cot 1 x Figure 5.3.11 Graph of y = cot 1 x Similarly, it can be shown that the inverse cosecant y = csc 1 x is a function whose domain is | x | ≥ 1 and whose range is π 2 y π
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Unformatted text preview: − π 2 ≤ y ≤ π 2 , y n= 0. Likewise, the inverse secant y = sec − 1 x is a function whose domain is | x |≥ 1 and whose range is 0 ≤ y ≤ π , y n= π 2 . csc − 1 (csc y ) = y for − π 2 ≤ y ≤ π 2 , y n= (5.10) csc (csc − 1 x ) = x for | x |≥ 1 (5.11) sec − 1 (sec y ) = y for 0 ≤ y ≤ π , y n= π 2 (5.12) sec (sec − 1 x ) = x for | x |≥ 1 (5.13) It is also common to call cot − 1 x , csc − 1 x , and sec − 1 x the arc cotangent , arc cosecant , and arc secant , respectively, of x . The graphs of y = csc − 1 x and y = sec − 1 x are shown in Figure 5.3.12:...
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