(32.) Again, taking the negative angle A C P"" - A, we have MP СР СР (33.) It will be easily seen that all the formulæ of this section hold for angles greater than 90°, or for negative angles, as well as for those which are positive and less than 90°. SECTION III. Trigonometrical Functions of the Sum and Difference of two Angles.— Formula resulting therefrom. THE formula which we shall now proceed to investigate are among the most important of Trigonometry. (34.) To find the sine of the sum or difference of two angles in terms of the sines and cosines of the angles themselves : Let the angle ACP = A, PCQ 1 B; then will A CQ=A+ B. With centre C and any radius CA describe the arc A PQ. Draw MP, KG, NQ perpendiculars to CA; QG perpendicular to C P, and F G to NQ; Then Fig. 1. KM A QN sin. (A + B) C Q M P СР C Q and the angles at F and M are right angles.. the triangles QFG, CMP are similar. Now QN NF + FQ KG + FQ; But by similar triangles Sin. (A+B) C P' G Fig. 2. N K sin. A cos. B + cos A. sin. B, the same form as that above deduced. In the same manner it may be shown in any other case in which A + B is less than 2π, that by virtue of the convention established in Art. (22.), the same formulæ will be true; and since all the trigonometrical functions of an angle are the same as for the angle increased by 2, this must hold for angles of all magnitudes. Sin. (AB) = sin. (π = (35.) To find the since sin. (A) and cos. (A) = two angles : The value of CN СР sin. (AB) Art. (25.) A + B) sin. (A). cos. B + cos. (π sin. A Art. (25.) - cos. A sin. B; cos. A Art. (26.) A) sin. B expression for the cosine of the sum or difference of (which cos. (A + B)) might be investigated in the QN same manner as that of or the sin. (A+B), or it may be simply C P' deduced from the above expression for this latter quantity, in the following (36.) These formulæ afford instances of the generalization effected by the use of the negative sign. The formule also for the difference of two angles is such as results from the substitution of B for B in the formulæ for the sum of those angles. Thus Sin. (AB) sin. (A + (- B)) cos. (A cos. A cos. (− B) — sin. A. sin. (― B) (37.) From what has now been stated, it appears that, with respect to the formulæ of this and the preceding section (which are the fundamental ones of the science), we may consider as proved the proposition in Art. (24.), which asserts that, by virtue of the convention there stated respecting the signs of lines, all formula which hold for angles less than 90°, hold equally for angles of any magnitude. Therefore, we conclude, by induction, that the rule will hold in all cases *. The explanation here given of this conventional use of the negative sign is such as will, we conceive, be most easily understood by the student in his earlier progress in mathematical investigations. The subject may, however, be placed in a somewhat different and more general point of view. Suppose a and b to represent the magnitudes of two known lines in the data of a geometrical problem to be solved algebraically; and let the magnitude of a line, measured in an assumed direction from a given point along an indefinite line given in position, be denoted by x, the object of the problem being the determination of the value of x; and suppose the resulting equation to be x = a − b. If a be greater than 6, the geometrical interpretation of this equation presents no possible difficulty; the magnitude of the required line must equal the difference of a and b, and the line must be measured in the direction first assumed. But let us suppose b greater than a; our result will become x = (b-a), and the question arises-what meaning attaches to this negative value of x? Does it indicate a geometrical impossibility, and admit of no geometrical interpretation; or does the circumstance of b being greater than a merely present a modification of our problem, still leaving it susceptible of a geometrical solution? The latter supposition is the true one. A general rule of interpretation may be established for all such cases accordant with the conventional use of the negative sign explained in the text, as indicating the directions of lines; consequently the equation x=- (ba) shows that the line represented by a must be measured in a direction opposite to that in which it must be measured when a is greater than b. The reasoning, however, on which the proof of the generality of elementary propositions of this nature depends, can hardly be fully appreciated by the student in his first advances in mathematics; and we have therefore thought it better to present the subject to him under the more simple rather than under the more general form, referring him for the latter to our treatise on the Application of Algebra to Geometry. The utility of this application of the negative sign will in all cases be similar to what we have shown it to be in trigonometry. (38.) By addition and subtraction the following formulæ are deduced immediately from those investigated in Art. (34.):— Sin. (A+B) + sin. (A – B) = 2 sin. A. cos. B sin. (A+B) sin. (A cos. (A B) B) B) 2 cos. A sin. B COS. cos. (A + B) (40.) By division, we have from the same formulæ, sin. A. cos. B + cos. A. sin. B Sin. (A+B) = sin. (A-B) sin. A. cos. B - cos. A sin. B sin. A sin. B (dividing numerator and denominator by cos. A. cos. B). Expanding the right-hand sides of these equations by the formulæ of Art. (34.), we have, by addition and subtraction, (42.) From these formulæ we obtain, by division, A B 2 2 A + B sin. 2 =tan. cos. A + cos. B A + B cot. 2 (43.) In formulæ, Art. (34.) let B = A |