Trigonometry

Trigonometry, originally the branch of geometry which had to do with the measurement of plane triangles. This gradually resolved itself into the investigation of the relations between the angles of the triangle, for the simple reason that all triangles having the same sets 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 right-angled 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 centre O describe the two circles PA and MQ. The appropriate measure of the angle at O is the ratio of the subtended arc to the radius —i.e. either $\backslash widehat\{AP\}/OP$ or $\backslash widehat\{MQ\}/OM$ (see CIRCLE). This measure we shall adopt throughout, and shall represent it by the symbol $\backslash theta$. If QN is drawn perpendicular 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 $\backslash theta$. Thus the ratio PM/OP or QN/OQ is the sine of $\backslash theta$. It is evidently half the chord of the angle $2\backslash theta$; and its value is numerically less than $\backslash theta$, because PM being less than the arc PA, the ratio PM/OM is the tangent of $\backslash theta$. MP is, in fact, the geometric 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 OQ/ON is called the secant of the angle $\backslash theta$. 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 $\backslash theta$, are called the cosine, cotangent, and cosecant of $\backslash theta$. For any given angle there are, then, six trigonometrical functions. It is obvious that these functions are mutually dependent. Indeed, if any one is given the other five can at once be calculated. For instance, the well-known relation $OM^2\; +\; MP^2\; =\; OP^2$ gives at once by dividing by $OP^2$

or, as it is usually written,

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 reciprocal of the tangent. The sine and cosine are never greater than unity, and the secant and cosecant are never less than unity. 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-clockwise. Then as the angle AOP increases from zero to a right angle the sine evidently grows from zero to unity; while at the same time the cosine diminishes from unity to zero. Continuing the increase so that AOP becomes an obtuse angle, we find that the sine begins to diminish, and that the cosine begins to increase numerically but towards the left of O. In other words, the cosine becomes negative, and continues so until 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, being thenceforward measured downwards until OP has made one complete revolution. After one complete revolution both sine and cosine, and also secant and cosecant, begin to go through exactly the same cycle of changes in magnitude and sign as at first. They are therefore periodic functions (see PERIOD), and their period is $2\backslash pi$ 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 sufficient in constructing tables of the trigonometrical functions to tabulate for angles from $0^\backslash circ$ to $90^\backslash circ$ inclusive. For example, the angle $130^\backslash circ$ ($90^\backslash circ\; +\; 40^\backslash circ$) has the same sine as the angle of $50^\backslash circ$ ($90^\backslash circ\; -\; 40^\backslash circ$); and its cosine differs only by being negative. Of greater practical importance than the tables of the functions themselves are the tables of their logarithms. These are generally tabulated for every degree and minute of angle from $0^\backslash circ$ to $90^\backslash circ$; and proportional parts are added by which is readily calculated the number corresponding to an angle involving seconds of arc.

The calculation of the functions and their logarithms is a sufficiently laborious task. It is generally effected by means of Series (q.v.), although the values for certain particular angles can be found by the simplest of arithmetical operations. Thus, the cosine of $60^\backslash circ$ is evidently $\backslash frac\{1\}\{2\}$; sine $60^\backslash circ$ is therefore $\backslash frac\{1\}\{2\}\backslash sqrt\{3\}$; tangent $60^\backslash circ$ is $\backslash sqrt\{3\}$; and so on. What might be called the fundamental series for the sine and cosine in terms of the arc are:

If we make all the signs in these two series positive we get two other functions of $\backslash theta$, which are called the hyperbolic sine and cosine of $\backslash theta$, and are written $\backslash sinh.\; \backslash theta$ and $\backslash cosh.\; \backslash theta$ respectively. Related to these functions there are the hyperbolic tangent, cotangent, secant, and cosecant; and they are connected by relations similar to, though not quite identical with, the ordinary circular functions. We may see, by adding the series with signs all positive, that the sum of the hyperbolic sine and cosine is the exponential of $\backslash theta$. Demoiivre's theorem gives the corresponding equation for the circular sine and cosine (see DEMOIVRE, and QUATERNIONS). The reason for the names circular and hyperbolic may be partially indicated thus: The relation $\backslash cos^2\; \backslash theta\; +\; \backslash sin^2\; \backslash theta\; =\; 1$ may be put in the form $x^2\; +\; y^2\; =\; a^2$, which is the equation of a circle of radius $a$, referred to rectangular axes (see GEOMETRY). The equation of the rectangular Hyperbola (q.v.) is $x^2\; -\; y^2\; =\; a^2$, to which there corresponds the relation $\backslash cosh^2\; \backslash theta\; -\; \backslash sinh^2\; \backslash theta\; =\; 1$. The hyperbolic sines and cosines are really exponential functions, and are not periodic. They are of constant occurrence both in the higher analysis and in mathematical physics. To facilitate their use in calculation, tables have recently been constructed.

Besides the series given above, there are many others, some of which are particularly serviceable for calculating the values of the functions or the values of their logarithms. There are also the converse series, by which an angle is found in terms of one of its circular functions. One of the simplest, and at the same time most historically famous, of these is Gregory's series, which expresses an angle in ascending powers of its tangent. It is as follows:

Of great importance are the addition formulæ which express any required function of the sum or difference of two given angles in terms of the trigonometrical functions of these angles. They are readily established for the circular functions by application of the elementary theorems of orthogonal projection. Similar formulæ hold for the hyperbolic functions. As plane trigonometry has to do chiefly with the solution of plane triangles, so spherical trigonometry is devoted to the discussion of spherical triangles. In navigation, geodesy, and astronomy the formulæ of spherical trigonometry are in constant use. The ordinary text-books on trigonometry do little more than present the subject in its practical bearings. An introductory study of the general theorems in analysis which have sprung out of the development of trigonometry will be found in Chrystal's Algebra (part ii.). It is impossible to make any progress in the higher mathematics without a thorough knowledge of the properties of the trigonometrical functions.