SECTION 7.1 The Inverse Sine, Cosine, and Tangent Functions 477 Figure 9 2p– 2 p– 2 u p 3p– 2 2p 2p 5p– 2 21 1 0 # u # p Look at Table 2 and Figure 9. The only angle θ within the interval π [ ] 0, whose cosine is 0 is π 2 . [Note that π cos 3 2 and π ( ) − cos 2 also equal 0, but they lie outside the interval π [ ] 0, , so these values are not allowed.] Therefore, π = − cos 0 2 1 θ cosθ 0 1 6 π 3 2 4 π 2 2 3 π 1 2 2 π 0 2 3 π 1 2 − 3 4 π 2 2 − 5 6 π 3 2 − π 1− Table 2 Finding the Exact Value of an Inverse Cosine Function Find the exact value of: − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ − cos 2 2 1 Solution EXAMPLE 5 Let θ = − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ − cos 2 2 . 1 Then θ is the angle, θ π ≤ ≤ 0 , whose cosine equals − 2 2 . θ θ π θ θ π = − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ ≤ ≤ =− ≤ ≤ − cos 2 2 0 cos 2 2 0 1 Look at Table 2 and Figure 10. The only angle θ within the interval π [ ] 0, whose cosine is − 2 2 is π3 4 , so π − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = − cos 2 2 3 4 1 Figure 10 p 1 2 2 2 3p 4 21 0 … u … p u Now Work PROBLEM 21 5 Define the Inverse Tangent Function Figure 11 shows the graph of = y x tan . Because every horizontal line intersects the graph infinitely many times, it follows that the tangent function is not one-to-one. Figure 11 y x x x tan , , = −∞< <∞ not equal to odd multiples of y 2 , π −∞< <∞ 2p– 2 p– 2 x p 3p ––– 2 2p 22p 2p 5p ––– 2 2 23p ––– 2 y 21 1 5p ––– 2 25p ––– 2 x 5 23p ––– 2 x 5 3p ––– 2 x 5 5p ––– 2 x 5 2p–– 2 x 5 p–– 2 x 5
RkJQdWJsaXNoZXIy NjM5ODQ=