Calculus: Analytic Geometry and Calculus, with Vectors

462 Trigonometric functions

The inverse cosine is the inverse of the function fl, with domain 0 5

x < it and range -1 < x <_ 1, for which fi(x) = cos x. Graphs of

x = cos i y, of y = cos-I x, and of the relevant parts of y = cos x and

x = cos y appear in Figures 8.355 and 8.356. When -1 < x < 1 and

Figure 8.355

y = cos-I x, then cos y = x, 0 < y < r, and sin y > 0. Again Theorem

8.33 implies that dy/dx exists, so (-sin y) (dy/dx) = 1 and

(8.357) dx cos-Ix =

dy

_ sinl -1 =

-1

Y 1 - cos2 y 1 - X2

This gives the integration formula

(8.358) 11 dx = -cos-I x -I- c (IxI < 1)

xf

2

which, because sin-' x + cos I x = 7r/2, does not contradict (8.354).

=tan-1 x

As Figure 8.171 may suggest, the in-

z= taa v Y verse tangentis theinverse ofth fe unc-

2 -------- tion fl, with domain -7r/2 < x < 7r/2

and range - oo < x < oo, for which

x fi(x) = tan x. The graph of y = tan-' x,

--- - 1 2 W ish h 'is t e same as a part (sometimesh

Figure 8.36 called the principal part) of the graph of

x = tan y, is shown in Figure 8.36. The

domain of the inverse tangent is the entire set of numbers, and if

y = tan-' x, then tan y = x and sec2 y(dy/dx) = 1, so

(8.361) tan-' x = ddx

x sec y 1 -I- tangy = 1 + x2

This gives the important formula

(8.362)

J

1-}1-xsdx=tan-Ix+ c.