1549901369-Elements_of_Real_Analysis__Denlinger_

432 Chapter 7 • The Riemann Integral

Theorem 7.7.31 The sine and cosine functions obey the following identities:

\ix,yEJR, (a) sin(x+y)=sinxcosy+cosxsiny;

(b) sin(~ - x) = cosx and cos(~ - x) = sin x;

(c) cos(x+y) =cosxcosy- sinxsiny.

Proof of (a): Let x, y be fixed real numbers, and let z = x + y. Then

\it E JR, :t[sintcos(z - t) + costsin(z -t)]

= (sint)[-sin(z - t)(-1)] + cos(z - t)(cost) + (cost)[-cos(z - t)] + sin(z -

t)(-sin t)

= o.

Thus, sin t cos(z - t) +cost sin(z - t) = K , a constant. ( 40)

Letting t = 0 in (40), we have 0 + 1 sinz = K. That is, K = sinz. Letting

t = x in (40), we have

sinxcos(z - x) + cosxsin(z - x) = sinz. (41)

Butz = x + y, so Equation (41) becomes

sin x cos y + cos x sin y = si n ( x + y).

The proofs of (b) and (c) are easy consequences of (a) and previous identities.

We stop with these identities, because from them and previous identities,

we can derive all the remaining trigonometric identities in the familiar manner.

Exercise 7. 7 .32 State and prove the derivative formulas for the remaining

trigonometric functions.

Exercise 7. 7 .33 State and prove t he integral ( antiderivative) formulas for all

six trigonometric functions.

Theorem 7.7.34 (Characterization of the Sine Function) The only

function F : JR __, JR such th at

(a) \ix E JR, F"(x) = -F(x),

(b) F(O) = 0, and

(c) F'(O) = 1

is the function F( x) = sin x.