Appendix III

(To § 158 and Chapter IX)

The circular functions

THE reader will find it an instructive exercise to work out the theory of the circular functions, starting from the definition

(1) y = y(x) = arctan x =0x dt 1 + t2. Df.115

The equation (1) defines a unique value of y corresponding to every real value of x. As y is continuous and strictly increasing, there is an inverse function x = x(y), also continuous and steadily increasing. We write

(2) x = x(y) = tan y. Df.

If we define π by the equation

(3) 1 2π =0 dt 1 + t2, Df.
then this function is defined for 1 2π < y < 1 2π.

We write further

(4) cos y = 1 1 + x2, sin y = x 1 + x2, Df.
where the square root is positive; and we define cos y and  sin y, when y is 1 2π or 1 2π, so that the functions shall remain continuous for those values of y. Finally we define cos y and  sin y, outside the interval [1 2π, 1 2π], by
(5) 2 tan(y + π) = tan y, cos(y + π) = cos y, sin(y + π) = sin y. Df.

We have thus defined cos y and  sin y for all values of y, and tan y for all values of y other than odd multiples of 1 2π. The cosine and sine are continuous for all values of y, the tangent except at the points where its definition fails.

The further development of the theory depends merely on the addition formulae. Write

x = x1 + x2 1 x1x2,
and transform the equation (1) by the substitution
t = x1 + u 1 x1u,u = t x1 1 + x1t.

We find

arctan x1 + x2 1 x1x2 =x1x2 du 1 + u2 =0x1 du 1 + u2 +0x2 du 1 + u2 = arctan x1 + arctan x2.

From this we deduce

(6) tan(y1 + y2) = tan y1 + tan y2 1 tan y1 tan y2,
an equation proved in the first instance only when y1y2, and y1 + y2 lie in [1 2π, 1 2π], but immediately extensible to all values of y1 and y2 by means of the equations (5).

From (4) and (6) we deduce

cos(y1 + y2) = ±(cos y1 cos y2 sin y1 sin y2).
To determine the sign put y2 = 0. The equation reduces to cos y1 = ± cos y1, which shows that the positive sign must be chosen for at least one value of y2, viz. y2 = 0. It follows from considerations of continuity that the positive sign must be chosen in all cases. The corresponding formula for sin(y1 + y2) may be deduced in a similar manner.

The formulae for differentiation of the circular functions may now be deduced in the ordinary way, and the power series derived from Taylor’s Theorem.

An alternative theory of the circular functions is based on the theory of infinite series. An account of this theory, in which, for example, cos x is defined by the equation

cos x = 1 x2 2! + x4 4!
will be found in Whittaker and Watson’s Modern Analysis (Appendix A).