SINE and We separate from the article TRIOONOUETItY the mere description and properties of these fundamental terms, which, though originally derived from simple trigonometry, are now among the most useful foundations of mathematical expresnion. For what we have to say on their history, we refer to the article just cited.
According to the ancient system of trigonometry, the eine and cosine are only names given to the abscissa and ordinate of a point, not with reference to the position of that point in space, but to the radius vector of that point and its angle. Thus, measuring angles from the ling o and in the direction of the arrow, the angle s o r has an infinite number of sines and cosines. With reference to the radius o r, r N is the sine and o N the cosine of L NOr; but with reference to the radius o q, q rt is the sine and o it the cosine. The fundamental relation (sine op + (cosine = is obvious enough.
The student. always began trigonometry with this multiplicity of definitions, and with the idea of come particular radius being necessary to the complete definition of the sine and cosine. But as he proceeded, he was always taught to suppose the radius a unit; that is, always to adopt that line as a radius which was agreed upon to be represented by 1. Hence he gradually learned to forget his first definition ; and, passing from geometry to arithmetic, to use the following: r o being unity, the sine of NOP is es, which is therefore in arithmetic the fraction which r y is of P 0 ; and the cosine is the fraction which 0 N is of P 0. If q 0 had been used as a unit, the result would have been the same; for by similar triangles, II q is the same fraction of Q o which N p is of P O.
In the most modern trigonometry, and for cogent reasons. the student is never for a moment allowed to imagine that the siuo and cosine are in any manner representatives of lines. In a practical point of view, the final definition of the old trigonometry coincides exactly with that of the new ; but the latter has this advantage, that all subsequent geometrical fortnulte are swell to be homogeneous in a much more distinct manner. The definition is this : The sine of N 0 P is not it is nor any number to represent. a r ; it is the fraction which N r is of r o, considered as an abstract number. Thus if 0 N, N P, r o, be in the pro
portion of 3, 4, and 5, P N is of 0 P : this is the sine of NO r, not of any line, nor any line considered as g of a unit; but simply 1, four fifths of an abstract unit, Similarly the cosine is the fraction which o N is of 0 is In just tho same manner the abstract number w, or 3'14159 ..., is not styled (as it used to be) the circumference of a circle whose diameter is a unit, but the proportion of the circumference to the diameter, the number of times which any circumference contains its diameter. We cannot too strongly recommend the universal adoption of this change of style, a slight matter with reference to mere calculation of results, but one of considerable importance to a correct understanding of the meaning of.formulit.
The line o e being considered as positive [SIGN), the signs of r N and. N 0 determine those of the sine and cosine; and tho manner in which the values of these functions are determined when the angle is nothing, or ono, two, or three right angles, is easy enough. The following short table embraces all the results of sign :— Read this na follows :—When the angle =0, the sine =0; front thence to a right angle the sine is positive : at the right angle the sine is + 1; from thence to two right angles the sine is positive, &c.
The fundamental theorems of the sine and cosine, from which all their properties may be derived, are, all which theorems are in fact contained in any one of tilCIO, su soon as that one is shown to be universally true. It frequently happens however that the student is allowed to assume the universal truth of these theorems upon too slight a foundation of previous proof : draw. ing a figure for instance in which both angles aro less than a right angle. We give, as an instance, the proof of the first formula when both angles are greater than two right angles. Let x 0 r= a, P o q=b, both angles being measured in the direction of revolution indicated by the arrow. The sum is four right angles+ x o q, which has the same sine and cosine as x o g. From any point g in o Q draw perpendiculars on o x and o r, and complete the figure as shown. Then sin (a + b) is positive, and is the fraction which g m is of g 0, or g M : Q o ; g M and g o being expressed in numbers. But