Calculus: Analytic Geometry and Calculus, with Vectors

442 Trigonometric functions

These formulas, which we have used many times, have at long last been

proved. For values of x for which the functions are defined, we obtain

(8.151) dxtan x =

dx cos x

(8.152) dxcotx

= dxs sx

(8.153) dxsec x =

dx cos x

(8.154) dxcsc x =

dx -sin x

cost x + sin2 x= sect x

cost x

• cos' x - sin2 x
  sin2 x
  sin x

= - csc x

= sec x tan x

cost x

• cos x
  sin2 x

= - csc x cot X.

The graphs of sin x and cos x are so important that we reproduce Fig-

ure 1.58 in Figure 8.16 and give further attention to the procedure by

Figure 8.16

which reasonably accurate graphs are quickly sketched with or without

use of graph paper. The trick is to sketch guide lines I unit above and

1 unit below the x axis, to hop 3 units and a bit more to the right of the

origin to mark r, to make another such hop to mark 21r, and then sketch

reasonably good copies of the figure. Each graph has slope l or -1 where

it crosses an axis, and noticeable contradictions of this fact should not

appear. In the problems at the end of the next section, we shall obtain

formulas from which sin 8 and cos 0 can, for a given 0, be calculated as

accurately as we wish. Meanwhile, we can be interested in Figure 8.17,

which enables us to obtain reasonable estimates of sin 8 and cos 0 when

0 < 0 < r/2. The circle has radius 1, and the radial lines make angles

0.1, 0.2, , 1.5 with the positive x axis. For example, the rough

approximations

sin 0.35

=0.14=0.34,

cos 0.35

=094=0.94,

034

tan 0.35 =

0.94 =

0.36

can be read from the figure.

With the aid of information about derivatives, it is easy to see that

the graph of the function t for which t(x) = tan x has the form shown in