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

Section 1.5 Inverse Functions and Logarithms 65
x
y
π
π
2
The inverse cosine function, cos 21 , has domain f21, 1g and range f0, g. Its graph is
shown in Figure 22.
The tangent function can be made one-to-one by restricting it to the interval
s2y2, y2d. Thus the inverse tangent function is defined as the inverse of the function
f sxd − tan x, 2y2 , x , y2. (See Figure 23.) It is denoted by tan 21 or arctan.
_1
FIGURE 22
y − cos 21 x − arccos x
y
0
1
x
tan 21 x − y &? tan y − x and 2 2 , y , 2
ExamplE 13 Simplify the expression cosstan 21 xd.
SOLUTION 1 Let y − tan 21 x. Then tan y − x and 2y2 , y , y2. We want to find
cos y but, since tan y is known, it is easier to find sec y first:
sec 2 y − 1 1 tan 2 y − 1 1 x 2
π
_ 2
π
0 x
2
sec y − s1 1 x 2
ssince sec y . 0 for 2y2 , y , y2d
figure 23
y − tan x, 2 2 , x , 2
y
FIGURE 24
œ„„„„„ 1+≈
1
x
Thus cosstan 21 xd − cos y − 1
sec y − 1
s1 1 x 2
SOLUTION 2 Instead of using trigonometric identities as in Solution 1, it is perhaps
easier to use a diagram. If y − tan 21 x, then tan y − x, and we can read from Figure 24
(which illustrates the case y . 0) that
1
cosstan 21 xd − cos y −
s1 1 x 2
The inverse tangent function, tan 21 − arctan, has domain R and range s2y2, y2d.
Its graph is shown in Figure 25.
■
y
π
2
0
x
FIGURE 25
y − tan 21 x − arctan x
_ π 2
We know that the lines x − 6y2 are vertical asymptotes of the graph of tan. Since
the graph of tan 21 is obtained by reflecting the graph of the restricted tangent function
about the line y − x, it follows that the lines y − y2 and y − 2y2 are horizontal
asymptotes of the graph of tan 21 .
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