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point, Q , to start from A on O X (cf. [fig.]27), and to rotate in the positive direction (anti-clockwise, in the figure considered) round the circumference of the circle for any number of times, finally resting at any point, e.g. at P or P ′ or P ″ or P ‴ . Then the total length of the curvilinear circular path traversed, divided by the radius of the circle, r , is the generalized definition of a positive angle of any size. Let x , y be the coordinates of the point in which the point Q rests, i.e. in one of the four alternative positions mentioned in [fig.]27; x and y (as here used) will either be x and y , or x ′ and y , or x ′ and y ′ , or x and y ′ .

Then the sign of this generalized angle is ${\displaystyle \frac{y}{r}}$ and its cosine is ${\displaystyle \frac{x}{r}}$. With these definitions the functional relations $v=\sin u$ and $w=\cos u$, are at last defined for all positive real values of $u$. For negative values of $u$ we simply take rotation of $Q$ in the opposite (clockwise) direction; but it is not worth our while to elaborate further on this point, now that the general method of procedure has been explained.

These functions of sine and cosine, as thus defined, enable us to deal with the problems concerning the triangle from which Trigonometry took its rise. But we are now in a position to relate Trigonometry to the wider idea of Periodicity of which the importance

was explained in the last chapter. It is easy to see that the functions $\sin u$ and $\cos u$ are periodic functions of $u$. For consider the position, $P$ (in [fig.]27), of a moving point, $Q$, which has started from $A$ and revolved round the circle. This position, $P$, marks the angles ${\displaystyle \frac{\text{arc }AP}{r}}$, and $2\pi +{\displaystyle \frac{\text{arc }AP}{r}}$, and $4\pi +{\displaystyle \frac{\text{arc }AP}{r}}$, and $6\pi +{\displaystyle \frac{\text{arc }AP}{r}}$, and so on indefinitely. Now, all these angles have the same sine and cosine, namely, ${\displaystyle \frac{y}{r}}$ and ${\displaystyle \frac{x}{r}}$. Hence it is easy to see that,

if $u$ be chosen to have any value, the arguments $u$ and $2\pi +u$, and $4\pi +u$, and $6\pi +u$, and $8\pi +u$ and so on indefinitely, have all the same values for the corresponding sines and cosines. In other words, {4} \sin u &= \sin(2\pi + u) &&= \sin(4\pi + u) &&= \sin(6\pi + u) &&= \text{etc.}; \ \cos u &= \cos(2\pi + u) &&= \cos(4\pi + u) &&= \cos(6\pi + u) &&= \text{etc.}

This fact is expressed by saying that $\sin u$ and $\cos u$ are periodic functions with their period equal to $2\pi$.

The graph of the function $y=\sin x$ (notice that we now abandon $v$ and $u$ for the more familiar $y$ and $x$) is shown in [fig.]28. We take on the axis of $x$ any arbitrary length at pleasure to represent the number $\pi$, and on the axis of $y$ any arbitrary length at pleasure to represent the number $1$. The numerical values of the sine and cosine can never exceed unity. The recurrence of the figure after periods of $2\pi$ will be noticed. This graph represents the simplest style of periodic function, out of which all others are constructed. The cosine gives nothing fundamentally different from the sine. For it is easy to prove that $\cos x=\sin (x+{\displaystyle \frac{\pi }{2}})$; hence it can be seen that the graph of $\cos x$ is simply [fig.]28 modified by

drawing the axis of $OY$ through the point on $OX$ marked ${\displaystyle \frac{\pi }{2}}$, instead of drawing it in its actual position on the figure.

It is easy to construct a `sine' function in 28 which the period has any assigned value $a$. For we have only to write $y=\sin \frac{2\pi x}{a},$ and then $\sin \frac{2\pi (x + a)}{a}

= \sin \left(\frac{2\pi x}{a} + 2\pi\right) = \sin \frac{2\pi x}{a}.$ Thus the period of this new function is now $a$. Let us now give a general definition of what