The circular functions

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

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

If we define $$\pi$$ by the equation then this function is defined for $$-\frac{1}{2}\pi < y < \frac{1}{2}\pi$$.

We write further where the square root is positive; and we define $$\cos y$$ and $$\sin y$$, when $$y$$ is $$-\frac{1}{2}\pi$$ or $$\frac{1}{2}\pi$$, so that the functions shall remain continuous for those values of $$y$$. Finally we define $$\cos y$$ and $$\sin y$$, outside the interval $${[-\frac{1}{2}\pi, \frac{1}{2}\pi]}$$, by

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 $$\frac{1}{2}\pi$$. 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 $x = \frac{x_{1} + x_{2}}{1 – x_{1}x_{2}},$ and transform the equation (1) by the substitution $t = \frac{x_{1} + u}{1 – x_{1}u},\quad u = \frac{t – x_{1}}{1 + x_{1}t}.$

We find \begin{aligned} \arctan \frac{x_{1} + x_{2}}{1 – x_{1}x_{2}} &= \int_{-x_{1}}^{x_{2}} \frac{du}{1 + u^{2}} = \int_{0}^{x_{1}} \frac{du}{1 + u^{2}} + \int_{0}^{x_{2}} \frac{du}{1 + u^{2}} \\ &= \arctan x_{1} + \arctan x_{2}.\end{aligned}

From this we deduce an equation proved in the first instance only when $$y_{1}$$$$y_{2}$$, and $$y_{1} + y_{2}$$ lie in $${[-\frac{1}{2}\pi, \frac{1}{2}\pi]}$$, but immediately extensible to all values of $$y_{1}$$ and $$y_{2}$$ by means of the equations (5).

From (4) and (6) we deduce $\cos(y_{1} + y_{2}) = \pm(\cos y_{1}\cos y_{2} – \sin y_{1}\sin y_{2}).$ To determine the sign put $$y_{2} = 0$$. The equation reduces to $$\cos y_{1} = \pm\cos y_{1}$$, which shows that the positive sign must be chosen for at least one value of $$y_{2}$$, viz. $$y_{2} = 0$$. It follows from considerations of continuity that the positive sign must be chosen in all cases. The corresponding formula for $$\sin(y_{1} + y_{2})$$ 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 $\cos x = 1 – \frac{x^{2}}{2!} + \frac{x^{4}}{4!} – \dots$ will be found in Whittaker and Watson’s Modern Analysis (Appendix A).