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Pre-Calc Exam Notes 122

Pre-Calc Exam Notes 122 - x x y − 1 1 − π 2 π 2 π 2...

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122 Chapter 5 Graphing and Inverse Functions §5.3 Example 5.14 illustrates an important point: sin 1 x should always be a number between π 2 and π 2 . If you get a number outside that range, then you made a mistake somewhere. This why in Example 1.27 in Section 1.5 we got sin 1 ( 0.682) = − 43 when using the a sin 1 button on a calculator. Instead of an angle between 0 and 360 (i.e. 0 to 2 π radians) we got an angle between 90 and 90 (i.e. π 2 to π 2 radians). In general, the graph of an inverse function f 1 is the reFection of the graph of f around the line y = x . The graph of y = sin 1 x is shown in ±igure 5.3.5. Notice the symmetry about the line y = x with the graph of y = sin
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Unformatted text preview: x . x y − 1 1 − π 2 π 2 π 2 1 − π 2 − 1 y = sin − 1 x y = sin x y = x Figure 5.3.5 Graph of y = sin − 1 x The inverse cosine function y = cos − 1 x (sometimes called the arc cosine and denoted by y = arccos x ) can be determined in a similar fashion. The function y = cos x is one-to-one over the interval [0, π ], as we see in the graph below: x y − 1 1 π 2 π − π 2 3 π 2 y = cos x Figure 5.3.6 y = cos x with x restricted to [0, π ] Thus, y = cos − 1 x is a function whose domain is the interval [ − 1,1] and whose range is the interval [0, π ]. In other words:...
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