Page:Encyclopædia Britannica, Ninth Edition, v. 23.djvu/583

PLANE.] exponential values of the sine and cosine, De Moivre's theorem, and a great number of other analytical properties of the trigonometrical functions are due to Euler, most of whose writings are to be found in the Memoirs of the St Petersburg Academy.

The preceding sketch has been mainly drawn from the following sources:—Cantor, ''Gesch. d. Math.; Hankel, Gesch. d. Math.; Marie, Hist. des sc. math.; Suter, Gesch. d. Math.; Klügel, Math. Wörterbuch''.

Imagine a straight line terminated at a fixed point O, and initially coincident with a fixed straight line OA, to revolve round O, and finally to take up any position OB. We shall suppose that, when this revolving straight line is turning in one direction, say that opposite to that in which the hands of a clock turn, it is describing a positive angle, and when it is turning in the other direction it is describing a negative angle.



Before finally taking up the position OB the straight line may have passed any number of times through the position OB, making any number of complete revolutions round in either direction. Each time that the straight line makes a complete revolution round we consider it to have described four right angles, taken with the positive or negative sign according to the direction in which it has revolved; thus, when it stops in the position OB, it may have revolved through any one of an infinite number of positive or negative angles any two of which differ from one another by a positive or negative multiple of four right angles, and all of which have the same bounding lines OA and OB. If OB is the final position of the revolving line, the smallest positive angle which can have been described is that described by the revolving line making more than one-half and less than the whole of a complete revolution, so that in this case we have a positive angle greater than two and less than four right angles. We have thus shown how we may conceive an angle not restricted to less than two right angles, but of any positive or negative magnitude, to be generated.

Two systems of numerical measurement of angular magnitudes are in ordinary use. For practical measurements the sexagesimal system is the one employed: the ninetieth part of a right angle is taken as the unit and is called a degree; the degree is divided into sixty equal parts called minutes; and the minute into sixty equal parts called seconds; angles smaller than a second are usually measured as decimals of a second, the "thirds," "fourths," &c., not being in ordinary use. In the common notation an angle, for example, of 120 degrees, 17 minutes, and 14·36 seconds is written 120° 17' 14"·36. The decimal system measurement of angles has never come into ordinary use. In analytical trigonometry the circular measure of an angle is employed. In this system the unit angle is the angle subtended at the centre of a circle by an arc equal in length to the radius. The constancy of this angle follows from the geometrical propositions—(1) the circumferences of different circles vary as their radii; (2) in the same circle angles at the centre are proportional to the arcs which subtend them. It thus follows that the unit mentioned above is an angle independent of the particular circle used in defining it. The constant ratio of the circumference of a circle to its diameter is a quantity incommensurable with unity, usually denoted by $$\pi$$. We shall indicate later on (p. 571 sq.) some of the methods which have been employed to approximate to the value of this quantity. Its value to 20 places is 3·14159265358979323846; its reciprocal to the same number of places is ·31830988618379067153. In circular measure every angle is measured by the ratio which it bears to the unit angle. Two right angles are measured by the quantity if, and, since the same angle is 180°, we see that the number of degrees in an angle of circular measure $$\theta$$ is obtained from the formula $$180\times\theta/\pi$$. The value of the unit of circular measure has been found to 41 places of decimals by Glaisher (Proc. London Math. Soc., vol. iv.): the value of $$\frac{1}{\pi}$$, from which the unit can be easily calculated, is given to 140 places of decimals in Grunert's Archiv [sic], vol. i., 1841. To 10 decimal places the value of the unit angle is 57° 17' 44"·8062470964. The unit of circular measure is too large to be convenient for practical purposes, but its use introduces a simplification into the series in analytical trigonometry, owing to the fact that the sine of an angle and the angle itself in this measure, when the magnitude of the angle is indefinitely diminished, are ultimately in a ratio of equality.

If a point moves from a position A to another position B on a straight line, it has described a length AB of the straight line. It is convenient to have a simple mode of indicating in which direction on the straight line the length AB has been described; this may be done by supposing that a point moving in one specified direction is describing a positive length, and when moving in the opposite direction a negative length. Thus, if a point moving from A to B is moving in the positive direction, we consider the length AB as positive; and, since a point moving from B to A is moving in the negative direction, we consider the length BA as negative. Hence any portion of an infinite straight line is considered to be positive or negative according to the direction in which we suppose this portion to be described by a moving point; which direction is the positive one is, of course, a matter of convention.

If perpendiculars AL, BM be drawn from two points A, B on any straight line, not necessarily in the same plane with AB, the length LM, taken with the positive or negative sign according to the convention as stated above, is called the projection of AB on the given straight line; the projection of BA being ML has the opposite sign to the projection of AB. If two points A, B be joined, by a number of lines in any manner, the algebraical sum of the projections of all these lines is LM, that is, the same as the projection of AB. Hence the sum of the projections of all the sides of any closed polygon, not necessarily plane, on any straight line, is zero. This principle of projections we shall apply below to obtain some of the most important propositions in trigonometry.

Let us now return to the conception of the generation of an angle as in fig. 1. Draw BOB at right angles to and equal to AA.



We shall suppose that the direction from A to A is the positive one for the straight line AOA, and that from B to B for BOB. Suppose OP of fixed length, equal to OA, and let PM, PN be drawn perpendicular to A'A, B'B; then OM and ON, taken with their proper signs, are the projections of OP on A'A and B'B. The ratio of the projection of OP on B'B to the absolute length of OP is dependent only on the magnitude of the angle POA, and is called the sine of that angle; the ratio of the projection of OP on A'A to the length OP is called the cosine of the angle POA. The ratio of the sine of an angle to its cosine is called the tangent of the angle, and that of the cosine to the sine the cotangent of the angle; the reciprocal of the cosine is called the secant, and that of the sine the cosecant of the angle. These functions of an angle of magnitude an are denoted by $$\sin\alpha$$, $$\cos\alpha$$, $$\tan\alpha$$, $$\cot\alpha$$, $$\sec\alpha$$, $$\text{cosec }\alpha$$ respectively. If any straight line RS be drawn parallel to OP, the projection of RS on either of the straight lines A'A, B'B can be easily seen to bear to RS the same ratios which the corresponding projections of OP bear to OP: thus, if $$\alpha$$ be the angle which RS makes with A'A, the projections of RS on A'A, B'B are $$RS\cos\alpha$$ and $$RS\sin\alpha$$ respectively, where RS denotes the absolute length RS. It must be observed that the line SR is to be considered as parallel not to OP but to OP'', and therefore makes an angle \pi+ a with A A; this is consistent with the fact that the projections of SR are of opposite sign to those of RS. By observing the signs of the projections of OP for the positions P, P, P', P of P we see that the sine and cosine of the angle POA are both positive; the sine of the angle P'OA is positive and its cosine is negative; both the sine and the cosine of the angle POA are negative; and the sine of the angle POA'' are negative and its cosine positive. If $$\alpha$$ be the numerical value of the smallest angle of which OP and OA are boundaries, we see that, since these straight lines also bound all the angles $$2n\pi + \alpha$$, where n is any positive or negative integer, the sines and cosines of all these angles are the same as the sine and cosine of $$\alpha$$. Hence the sine of any angle $$2n\pi + \alpha$$ is positive if $$\alpha$$ is between 0 and $$\pi$$ and negative if $$\alpha$$ is between $$\pi$$ and $$2\pi$$, and the cosine of the same angle is positive if $$\alpha$$ is between 0 and $$\frac{1}{2}\pi$$ or $$\frac{3}{2}\pi$$ and $$2\pi$$ and negative if $$\alpha$$ is between $$\frac{1}{2}\pi$$ and $$\frac{3}{2}\pi$$.

In fig. 2 if the angle POA is $$\alpha$$, the angle $$POA$$ is $$-\alpha$$, P'OA is $$\pi - \alpha$$, POA is $$\pi + \alpha$$, POB is $$\frac{\pi}{2} - \alpha$$. By observing the signs of the projections we see that

,

,

.

Also

,

.

From these equations we have $$\tan(-\alpha) = -\tan\alpha$$, $$\tan(\pi - \alpha) = -\tan\alpha$$, $$\tan(\pi + \alpha) = -\tan\alpha$$, $$\tan(\frac{1}{2}\pi - \alpha) = \cot\alpha$$, $$\tan(\frac{1}{2}\pi + \alpha) = -\cot\alpha$$, with corresponding equations for the cotangent.

The only angles for which the projection of OP on B'B is the same as for the given angle POA ($$=\alpha$$) are the two sets of angles bounded by OP, OA, and OP', OA; these angles are $$2n\pi + \alpha$$ and $$2n\pi + \overline{\pi - \alpha}$$, and are all included in the formula $$r\pi + (-1)^r\alpha$$, where r is any integer; this therefore is the formula for all angles having the same sine as $$\alpha$$. The only angles which have the same cosine as an are those bounded by OA, OP and OA, OP'', and these are all included in the formula $$2n\pi\pm \alpha$$. Similarly it can be shown