TRIGONOMETRY, originally the branch of geometry which had to do with the measurement of plain triangles. This gradually resolved itself into the in vestigation of the relations between the angles of the triangle, for the simple reason that all triangles having the same set of angles are similar, so that if, in addition, one side is given the other two at once follow. It is easy to show from the Sixth Book of Euclid that, if we fix the values of the angles of a triangle, the ratio of the sides containing any one of these angles is the same whatever be the size of the triangle. This ratio is a definite function of the angles; and it is with the properties of such ratios that trigonometry has now to deal. The fundamental ratios are obtained from a triangle, of which one angle is the angle under consideration. It will suffice to show what these ratios are and how they have received their names. Let POM be the angle considered, PM being drawn perpendicular to OM. With center O describe the two circles PA and MQ. The appropriate measure of the angle at 0 is the ratio of the subtended arc to the radius—i. e., either AP/OP or MQ/OM (see CIRCLE). This measure we shall adopt throughout, and shall represent it by the symbol 0. If QN is drawn per pendicular to OM, then the ratio of any pair of sides of the triangle OQN is equal to the ratio of the corresponding sides of triangle OPM. All the possible ratios which can be formed are the so-called trigonometrical or circular functions of the angle 0. Thus the ratio PM/OP or QN/OQ is the sine of 0. It is evidently half the chord of the angle 2/0; and its value is numerically less than 0, because PM being less than the chord PA is less than the arc PA. Again, the ratio PM/ OM is the tangent of 0. MP is, in fact, the geometrical tangent drawn from the one extremity of the arc MQ till it meets the radius through the other extremity. For a similar reason the ratio OP/OM or 0Q/ON is called the secant of the angle O. In the same way the ratios OM/OP, OM/PM, OP/PM are respectively the sine, tangent, and secant of the angle OPM, which is the complement of the angle POM. Hence these ratios, regarded as functions of 19, are called the cosine, cotangent, and cosecant of 0. For any given angle there are, then, six trigono metrical functions. It is obvious that these functions are mutually dependent.
Indeed, if any one is given the other five can be at once calculated. For the well-known relation gives at once by dividing by OP' (sin. + (cos. or, as it is usually written, sin' 0 + cos' Then, again, the cosecant is the reciprocal of the sine, and the secant of the cosine. The tangent is the ratio of the sine to the cosine; and the cotangent is the recipro cal of the tangent. The sine and cosine are never greater than unity, and the se cant and cosecant are never less than uni ty. The tangent is less or greater than unity according as the angle POM is less or greater than half a right angle.
Suppose OP to rotate counter-clock wise. Then as the angle AOP increases from zero to a right angle the sine evi dently grows from zero to unity; while at the same time the cosine diminishes from unity to zero. Continuing the in crease so that AOP becomes an obtuse angle, we find that the sine begins to di minish, and that the cosine begins to increase numerically but toward the left of 0. In other words, the cosine becomes negative, and continues so till OP has completed three right angles. In the same way, as AOP passes through the value of two right angles and becomes re-entrant, the sine becomes negative, be ing thenceforward measured downward tilt OP has made one complete revolu tion. After one complete revolution both sine and cosine, and also secant and co secant, begin to go through exactly the same cycle of changes in magnitude and sign as at first. There are, therefore, periodic functions, and their period is or four right angles. The tangent and cotangent, however, go through their cycle of changes in half this period or two right angles. All possible numerical values of the functions are obtained in the first quadrant. It is therefore suf ficient in constructing tables of the trig onometrical functions to tabulate for an gles from 0° to 90° inclusive. For exam ple, the angle 130° (90°--F40°) has the same sine as the angle of 50 (90°--40°) ; and its cosine differs only by being nega tive. Of greater practical importance than the tables of the functions them selves are the table of their logarithms. These are generally tabulated for every degree and minute of angle from 0° to 90°; and proportional parts are added by which is readily calculated the num ber corresponding to an angle involving seconds of arc.