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Pre-Calc Exam Notes 123

Pre-Calc Exam Notes 123 - Inverse Trigonometric Functions...

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Inverse Trigonometric Functions Section 5.3 123 cos 1 (cos y ) = y for 0 y π (5.4) cos (cos 1 x ) = x for 1 x 1 (5.5) The graph of y = cos 1 x is shown below in Figure 5.3.7. Notice the symmetry about the line y = x with the graph of y = cos x . x y 0 1 1 π π 2 1 π π 2 1 y = cos 1 x y = cos x y = x Figure 5.3.7 Graph of y = cos 1 x Example5.15 Find cos 1 ( cos π 3 ) . Solution: Since 0 π 3 π , we know that cos 1 ( cos π 3 ) = π 3 , by formula (5.4). Example5.16 Find cos 1 ( cos 4 π 3 ) . Solution: Since 4 π 3 > π , we can not use formula (5.4). But we know that cos 4 π 3 = − 1 2 . Thus, cos 1 ( cos 4 π 3 ) = cos 1 ( 1 2 ) is, by definition, the angle y such that 0 y π and cos y =− 1 2 . That angle is y = 2 π 3 (i.e. 120 ). Thus, cos 1 ( cos 4 π 3 ) = 2 π 3 . Examples 5.14 and 5.16 may be confusing, since they seem to violate the general rule for
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