Calculus: Analytic Geometry and Calculus, with Vectors

478 Trigonometric functions

of quotients of polynomials in z. We suppose that -7r < 0 < 7r,
that -n/2 < 0/2 < a/2, and define z in terms of B by the formula

(8.51) z = tan 2

Differentiation gives

2

dB

Zsect 2^2

//

(1 + tang 0= 2

z,

and hence

(8.52) do = 1+z2dz.

Basic trigonometric identities and (8.51) give

2 tan

B B 2z

(8.531) sing=2sin2cos2= -1+52'

sect

2

(8.532) cos 0 = cost 2 - sin2 2 = 2 cos'2 - 1

2 2 1 - z2

-1=1+52-1=1+z2

sect^10

and, except when 0 is it/2 or -ir/2,

2 tan 2

2z

(8.533) tan 0 = =

1 - tang 2

1 - z2

so

In fact, Figures 8.541 and 8.542 enable us to write the six trigonometric

functions of 8/2 and 0 in terms of z. Thus each of sin 0, cos 0, tan 0,

1

Figure 8.541

1-z2

Figure 8.542

cot 0, sec 0, csc 0, and dB/dz is a quotient of polynomials in z. It follows

from this that if P and Q are polynomials in the six trigonometric func-

tions of x, then

f

PQ11