文档介绍：Section
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Inverse Trigonometric Functions
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In this section we will introduce the inverse trigonometric functions. We will begin with
the inverse tangent function since, as indicated in Section , we need it plete the
story of the integration of rational functions.
Strictly speaking, the tangent function does not have an inverse. Recall that in order
for a function f to have an inverse function, for every y in the range of f there must be
exactly one x in the domain of f such that f(x) = y. This is false for the tangent function
since, for example, both tan(0) = 0 and tan(π) = 0. In fact, since the tangent function is
periodic with period π, if tan(x) = y, then tan(x + nπ) = y for any integer n. However,
−ππ
the tangent function is increasing on the interval 2 , 2 , taking on every value in its
range (−∞, ∞) exactly once. Hence we may define
an inverse for the tangent function
if we consider it with the restricted domain −π, π. That is, we will define an inverse
2 2
tangent function so that it takes on only values in −π, π.
2 2
Definition The arc tangent function, with value at x denoted by either arctan(x) or
tan−1(x), is the inverse of the tangent function with restricted domain −π, π.
2 2
−ππ
In other words, for 2 < y < 2 ,
y = tan−1(x) if and only if tan(y) = x. ()
−1 −1 π−1 −−π
For example, tan (0) = 0, tan (1) = 4 , and tan ( 1) = 4 . In particular, note that
−1 −ππ
even though tan(π) = 0, tan (0) = 0 since 0 is between 2 and 2 , but π is not between
−ππ
2 and 2 .
The domain of the arc tangent function is (−∞, ∞), the range of the tangent function,
−ππ
and the range of the arc tangent function is 2 , 2 , the domain of the restricted tangent
function. Moreover, since
lim tan(x) = ∞
→π−
x 2
and
lim tan(x) = −∞,
→−π+
x 2
we have
π
lim tan−1(x) = ()
x→∞ 2
and
π
lim tan−1