Calculus: Analytic Geometry and Calculus, with Vectors

458 Trigonometric functions

for each n = 1, 2, 3,. It is true (and is easy to surmise) that, for each x,
JxIn/n! --> 0 as n -> co. Therefore,

x3 x5 x7

x2 x4 x6

cosx=1-2!+,i-6i+

9 The formulas at the end of the preceding problem have been proved to

be correct when x > 0. Use this fact to prove that the formulas are also correct

when x <--_ 0. Hint: Look first at the case in which x = 0. Then use the facts

that sin (-x) = - sin x and cos (-x) = cos x.

10 Have a good look at the formulas at the end of Problem 8 and start learning

them by obtaining some of the following results correct to four or more decimal

places.

sin 0.01 = 0.00999 9833

sin 0.02 = 0.01999 8667

sin 0.10 = 0.09983 3417

sin 0.20 = 0.19866 9331

sin 1.00 = 0.84147 0985

sin 1.10 = 0.89120 7360

cos 0.01 = 0.99995 0000

cos 0.02 = 0.99980 0007

cos 0.10 = 0.995C0 4165

cos 0.20 = 0.98006 6587

cos 1.00 = 0.54030 2306

cos 1.10 = 0.45359 6121

11 Digest the following idea. We have a desk calculator and National

Bureau of Standards Tables giving the values

sin 0.2345 = 0.23235 6699

cos 0.2345 = 0.97263 0641

If we want to find sin 0.23456 789 correct to eight decimal places, we can use

the identity

sin (x + 0.2345) = sin x cos 0.2345 + cos x sin 0.2345,

where x = 0.00006789. The values of sin x and cos x can easily be found correct

to 10 decimal places from the formulas at the end of Problem 8.

8.3 Inverse trigonometric functions Before coming to the an-

nounced subject of this section, we think about a general situation appli-

cations of which appear in many branches of mathematics. Suppose

we have an operator or transformer or mapper or function f that carries

or transforms or maps each element x of a set D (the domain of f) into

an element y of a set R (the range of f). In some cases the transformer

transforms two or more different elements of -D into the same element

of R. For example, if f is one of the six trigonometric functions and

f (x) exists, then f (x + 2a) = f (x) and hence more than one element of

the domain is carried into the same element f(x) of the range. When

problems involving domains and ranges are involved, it is very often

possible to eliminate confusion by singling out for special attention a

subset Dl of D such that to each y in R there corresponds exactly one x