third quadrant; and lastly, becomes negative again, and decreases from infinity to nothing, in the fourth quadrant. 15. These conclusions admit of a ready confirmation; and others may be deduced, by means of the analytical expressions in arts. 4 and 12. Thus, if A be supposed equal to 10, in equa. v, it will become. cos (B) = cos. O cos. B sin. (B) = sin.. cos. B sin. sin. B, sin. B. cos. 0. O =0: so that the above equations will become cos (OB) = sin. B. sin. (O+B) COS B. From which it is obvious, that if the sine and cosine of an arc, less than a quadrant, be regarded as positive, the cosine of an arc greater than 0 and less than will be negative, but its sine positive. If в also be made = O; then shall we have cos. O = · 1; sin 0 = 0. Suppose next, that in the equa. v, A = 0; then shall we obtain. which indicates, that every arc comprised between and 20, or that terminates in the third quadrant, will have its sine and its cosine both negative. In this case too, when BO, or the arc terminates at the end of the third quadrant, we shall have coss = 0, sin. † 0 = − 1. Lastly the case remains to be considered in which A = O or in which the arc terminates in the fourth quadrant. Here the primitive equations (V) give COS. (B) = ± sin. B sin. (B) = COS. B; so that in all arcs between O and O, the cosines are positive and the sines negative. 16. The changes of the tangents, with regard to positive and negative, may be traced by the application of the preceding results to the algebraic expression for the tangent; viz. sin. tan. For it is hence manifest, that when the sine and COS cosine are either both positive or both negative, the tangent will be positive; which will be the case in the first and third quadrants. But when the sine and cosine have different signs, the tangents will be negative, as in the second and fourth quadrants. The algebraic expression for the cotan COS. gent, viz. cot. ——, will produce exactly the same results. sin. The 1 The expressions for the secants and cosecants, viz. sec. = 1 -, cosec. = show, that the signs of the secants are the COS. sin, same as those of the cosines; and those of the cosecants the same as those of the sines. The magnitude of the tangent at the end of the first and third quadrants will be infinite; because in those places the sine is equal to radius, the cosine equal to zero, and therefore =∞(infinity). Of these, however, the former will be reckoned possitive, the latter negative. sin. COS. 17. The magnitudes of the cotangents, secants, and cosecants may be traced in like manner; and the results of the 13th, 14th, and 15th articles, recapitulated and tabulated as below. Sd. 7th. 11th. 15th. 4th. 8th. 12th. 16th. quadrants. + (VII.) We have been thus particular in tracing the mutations, both with regard to value and algebraic signs, of the principal trigonometrical quantities, because a knowledge of them is absolutely necessary in the application of trigonometry to the solution of equations, and to various astronomical and physical problems. 18. We may now proceed to the investigation of other expressions relating to the sums, differences, multiples, &c of arcs; and in order that these expressions may have the more generality, give to the radius any value в instead of confining it to unity. This indeed may always be done in an expression, however complex, by merely rendering all the terms homogeneous; that is, by multiplying each term by such a power of R as shall make it of the same dimension, as the term in the equation which has the highest dimension. the expression for a triple arc. Thus, sin, sin. 3A 3 sin. A-4 sin3. A(radius or sin. 3A R 2 = 1) Hence then, if consistently with this precept, n be placed for a denominator of the second member of each equation v (art. 12), and if ▲ be supposed equal to B, we shall have sin. A. COS. A 4 sin. A cos. A sin. (A + A) = And, in like manner, by supposing B to become successively equal to 2A, 3A, 4A, &c. there will arise And, by similar processes, the second of the equations just referred to, namely, that for cos. (A + B), will give suc cessively, 19. If, in the expressions for the successive multiples of the sines, the values of the several cosines in terms of the sines were substituted for them; and a like process were adopted with regard to the multiples of the cosines, other expressions would be obtained, in which the multiple sines would be expressed in terms of the radius and sine, and the multiple cosines in terms of the radius and cosines. Add together the expanded expressions for sin. (B + A), sin (B add to A), that is, sin. (B + A) = sin. B. cos. A + cos. B. sin. A, sin. (B A) sin. B. cos. A - cos. B. sin. A ; there results sin (B + A) + sin. (BA) = 2 cos. A. sin B: whence, sin. (BA) = 2 cos A. sin. B sin(B — A). Thus again, by adding together the expressions for cos (B+A) whence, cos. (B + A) = 2 cos. A . COS. B COS. (BA). Substituting in these expressions for the sine and cosine of BA, the successive values A, 2A, 3A, &c. instead of B; the following series will be produced. Several other expressions for the sines and cosines of multiple arcs, might readily be found: but the above are the most useful and commodious. 20 From the equation sín. 2a = it will be easy, when the sine of an arc is known, to find that of its half. For, substituting for cos A its value (R2 sin2 A), 2 sin A(R2-sin2 A) Here taking sin a for the unknown quantity, we have a quad Here we have omitted the powers of R that were necessary to ren der all the terms homologous, merely that the expressions might be brought in upon the page; but they may easily be supplied, when needed, by the rule in art. 18. ratic ratic equation, which solved after the usual manner, gives sin A±√{R2± } R√R2 sin2 2A If we make 2A = A'. then will a the last equation becomes A', and consequently, sin A' = ± √ R 2 ± ÷R √ R2 – Sin2 A' 2 (XII.) or sin A' = ± ÷ √2R2 ± 2R cos A': by putting cos A' for its value R2 - sin2 A multiplying the quantities under the radical by 4, and dividing the whole second number by 2. Both these expressions for the sine of half an arc or angle will be of use to us as we proceed. 21. If the values of sin (A + B) and sine (AB), given by equa. v, be added together, there will result R sin (A—B). (XIII.) sin ▲ . Cos B = R Sin (A + B ) + Also, taking sin (AB) from sin (A + B) gives sin B -- cos a =‡r sin (A+B)—4R . Sin (A—B) . . (XIV.) When A = B. both equa. xII and XIV, become cos A sin A = R sin 2A (XV.) 22. In like manner, by adding together the primitive expressions for cos (A + B), COS (a B), there will arise 2 cos A. cos B R ; whence, COS A COS B = R COS (A+B) + R. COS (A-B) (XVI.) And here, when A = B, recollecting that when the arc is nothing the cosine is equal to radius, we shall have COS ARCOS 2A + R2 . (XVII) 23. Deducting cos (A+B) from cos (AB), there will remain 2 sin A. sin B ; whence, R COS (A - B) COS (A + B) sin A . Sin B=1R. COS (A — B)—R. COS (A+B) (XVIII.) When AB, this formula becomes sin2 A = +R2 —R. COS 2A.. (XIX.) 24. Multiplying together the expressions for sin (A + B) and sin (AB), equa. v, and reducing, there results sin (A+B). sin (A — B) = sin3 A - sin2 B. And, in like manner, multiplying together the values of cos (A+B) and cos (A B), there is produced cos (A+B). COS (A — B) Cos2 A- COS B. sin B, is equal to (sin A+ sin B) X sin B), that is, to the rectangle of the sum and dif Here, since sin A (sin A →→→ ference |