James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

64 Chapter 1 Functions and Models
Since the definition of an inverse function says that
we have
f 21 sxd − y &? f syd − x
sin 21 x − y &? sin y − x and 2 2 < y < 2
sin 21 x ± 1
sin x
Thus, if 21 < x < 1, sin 21 x is the number between 2y2 and y2 whose sine is x.
ExamplE 12 Evaluate (a) sin 21 s 1 2d and (b) tansarcsin 1 3 d.
SOLUTION
(a) We have
sin 21 s 1 2d − 6
¨
figure 19
3
2 œ„2
1
because sinsy6d − 1 2 and y6 lies between 2y2 and y2.
(b) Let − arcsin 1 3 , so sin − 1 3 . Then we can draw a right triangle with angle as
in Figure 19 and deduce from the Pythagorean Theorem that the third side has length
s9 2 1 − 2s2 . This enables us to read from the triangle that
tansarcsin 1 3 d − tan − 1
2s2
■
y
The cancellation equations for inverse functions become, in this case,
π
2
sin 21 ssin xd − x for 2 2 < x < 2
_1
0
1
x
sinssin 21 xd − x for 21 < x < 1
_ π 2
figure 20
y − sin 21 x − arcsin x
y
The inverse sine function, sin 21 , has domain f21, 1g and range f2y2, y2g, and
its graph, shown in Figure 20, is obtained from that of the restricted sine function (Figure
18) by reflection about the line y − x.
The inverse cosine function is handled similarly. The restricted cosine function
f sxd − cos x, 0 < x < , is one-to-one (see Figure 21) and so it has an inverse function
denoted by cos 21 or arccos. y
π
1
0
π
2
π
x
The cancellation equations are
_1
0
1
cos 21 x − y &? cos y − x and 0 < y <
π
2
x
FIGURE 21
y − cos x, 0 < x <
cos 21 scos xd − x for 0 < x <
cosscos 21 xd − x for 21 < x < 1
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