Calculus: Analytic Geometry and Calculus, with Vectors

8.3 Inverse trigonometric functions 463

The inverse cotangent is the inverse of the function fl, with domain
0 < x < 7r and range - oo < y < oo. A graph of y = cot-1 x is shown

in Figure 8.37. The domain of the inverse cotangent is the entire set of

Figure 8.37

real numbers and if y = cot-1 x, then cot y = x and - csc2 y(dy/dx) = 1,

so

(8.371) d cot-1 x =

dy= -1 = -1 _ -1

dx dx csc2 y 1 + cot2 y 1 + x2'

As Figure 8.233 indicates, the graph of y = sec x presents a difficulty

that has not previously appeared. It is impossible to select an interval

D, of values of x such that sec x is defined and continuous over D, and

each y in the range of sec x is attained for some x in D1. The best we

can do is let D, be the interval 0 S x <-- 7r with the center point x = 7r/2

omitted. The inverse secant is then

the inverse of the function f1 with ly y=sec-lx

domain D, for which fl(x) = sec x. IT

The graph of y = sec-1 x, which ]P

x=secy

coincides with a part of the graph of ; 2

x = sec y, is shown in Figure 8.38. -1 1 x

If x > 1 or x < -1 and y = see-' x, Figure 8.38

then sec y = x, 0 < y < 7r, y o 7r/2,

and sec y tan y > 0. Again Theorem 8.33 implies that dy/dx exists, so

secytanydXI

and

(8.381) dxsec-1 x = dx =

secyltany secyIljtanyj

1 1

sec yJ sec2 -y- 1 IxI x2 - 1

This gives the formula

(8.382) r 1 dx = sec71 x + c,

.l IxI x2 - 1

which is valid when x > 1 and also when x < -1.