Calculus: Analytic Geometry and Calculus, with Vectors

460 Trigonometric functions

fl has an inverse defined over R. Since we must prove that fl' is differ-
entiable, we introduce difference quotients for fi'. Supposing that y
and y + Ay belong to the domain R of f,-', we define x and x + Ax by
the formulas x = fi'(y) and x + Ox = fi'(y + Ay) and obtain y = fi(x),

y + Ay = fi(x + .x). Since fl has a positive (or negative) derivative,

we conclude that Ox 54 0 when [1y 0 and that Ax 0 as Ay --p 0.

Therefore,

(8.332) limf i'(Y + AY) - f- 11(y) = lim

Ax

AY-0 AY ou-.0f (x + Ox) - fl(x)

• AUmofl(xi + Ax) - fl(x) fi(x) fi(fll(Y))
  Ox

This proves Theorem 8.33. Of course it is possible to abbreviate the

calculations by writing

(8.333)

dx

= lim - =

1 = 1 ,

dy °y-'0AY lim D

dy

A.,_0 Ax dx

but this one line is not, by itself, the equivalent of a theorem and proof

which present conditions under which the formula is correct.

Our general discussion of inverse functions and Theorem 8.33 will now

be used to guide us to six functions that are called inverse trigonometric

functions even though they are in fact inverses of restrictions of trigono-

metric functions. It will turn out that the six functions will all have

values between 0 and it/2, and that all of the information implied by

Figures 8.341, 8.342, and 8.343 will be correct, provided 0 < x < 1 in

2-i

Figure 8.341 Figure 8.342 Figure 8.343

Figure 8.341, x > 0 in Figure 8.342, and x > 1 in Figure 8.343. The

first figure shows, for example, that

X

sin (sin-1 x) = x, tan (sin-' x) =

1 __X 2

when 0 < x < 1. These triangles can sometimes provide helpful infor-

mation even when the functions appear in problems involving the mis-

fortunes of gamblers who bet on horse races.

To begin, let f be the 'trigonometric function for which f (x) = sin x,

the domain D being the infinite interval - oo < x < -o and the range