in the fourth quadrant, since the sine is then negative and the cosine positive. The tangent, therefore, changes its sign at the end of every quadrant, where it passes alternately through nothing and infinity; these being, indeed, the algebraic indications of the changes of sign. If the arc is regarded as negative, the rule for the tangents becomes inverted; the tangent being then negative in the first and third, positive in the second and fourth arcs. 6. For the cotangents the rule of the signs is the same as for the tangents: this is evident, because 7. So again, the rule for the secants is the same as for 1 the cosines; because sec = COS 8. The rule for the cosecants is, also, the same as for the sines; because cosec = sin 9. Proceeding in this way, a second, a third, &c. time over the circumference of the circle, like mutations would occur. The results of the whole may be thus tabulated : 10. Let it be remembered that when the tangent passes through infinity, the secant does, and for the same reason, viz. because they in that case could only limit each other at an infinite distance. The principal changes, then, in point of magnitude, may be easily traced, and tabulated as below; where the sign ∞ denotes infinity. 0°....90°.... 180°.... 270°.... 360°) 11. These particulars premised, the properties and relations of the sines, tangents, &c. of combined and multiple arcs, as well as rules for the solutions of triangles, may be investigated analytically. Let ABC, in the annexed figure, be any plane triangle, c the vertical angle, CD a perpendicular let fall from it upon the base or base produced, and let a, b, and c, denote the sides respectively opposite to the angles A, B, and C. = с B Then, since AC b, AD is the cosine of a to that radius; consequently, when radius is unity we have AD = b cos a. In like manner BD a cos B. A There If one fore, AD + BD = AB = c = a cos B + b cos a. of the angles, as A, were obtuse, the result would, notwithstanding, be the same; because, while on the one hand cos A would be negative, AD, lying on the contrary side of a to what it does in the figure, must be deducted from BD to leave AB, and a negative quantity subtracted, is equivalent to a positive quantity added. By letting fall perpendiculars from the angles A and B, upon the opposite sides, or their continuations, precisely analogous results will be obtained. They may be placed together, thus a = b cos c + c cos B ca cos B + b cos A (1.) 12. If BC, or a, be regarded as radius, CD will be the sine of the angle в to that radius; therefore, to the radius unity, CD will be a sin B. So again, for a like reason, CD = 6 sin A. Consequently, a sin в = b sin a, tors, the relations of all the six quantities may be thus expressed, These equations are, manifestly, of similar import with chap. ii. prop. 14. 13. From the last equations we have, Substituting these values of a and b for them in the sin c = sin a COS B+ sin в COS A. Now, since in every plane triangle, the sum of the three angles is equal to two right angles, A + B = supplement of c; and, since an angle and its supplement have the same sine, it follows that sin (A + B) = sin c; whence sin (A + B): sin A COS B + sin B COS A. 14. If in this equation в be regarded as subtractive, then will sin B obviously be subtractive also; but cos B will not change its sign, because it will still continue to be estimated in the same direction on the same radius The equation will, therefore, become sin (AB) sin A COS B sin B COS A. 15. Conceive B' to be the complement of в, and O to be the quarter of the circumference, or the measure of a right angle: then will B'○ —B, sin B′ = cos B, and cos B'sin B. But, by the preceding article, sin (A - B') sin A cos B' -sin B'cos A. Substituting for sin B', cos B', their values, there will result sin (A — B') COS A COS B. Hence, because B′ sin A sin B 40 - B, we have sin (A — B′) == = sin (A + B sin [(A+B)] sin [O (A + B) = COS (A+B)]. This value of sin (A - B′) being substituted for it in the equation above, it becomes COS A COS B sin A sin B. COS (A + B) 16. If в in this latter equation be made subtractive, sin B will become sin B, while cos B will not change (art. 3, 4). The equation will consequently be transformed into this: viz. COS (AB) COS A COS B + sin A sin B. If A and B be regarded as arcs instead of the angles which they measure, the results will be equally conclusive and correct. They may be expressed generally for the sines and cosines of the sums or differences of any two arcs or angles, by these two equations, viz. ^} (c.) sin (AB) = sin A COS B± sin в COS A COS (AB) COS A COS B sin A sin B 17. The actual value of the sine, cosine, &c. depends, obviously, not only upon the magnitude of the arc, but also upon that of the assumed radius. In the preceding investigation we have supposed it to be unity. If we wish to make the above or any other formulæ, applicable to cases where the radius has another value R, we for sin A, have only to substitute in the expression, COS A R tan A R sin A R for cos A, rally, we must so distribute the several powers of R, as to make all the terms homogeneous, as to the number of lines multiplied: this is effected by multiplying each term by such a power of R as shall make it of the same dimension, as that term in the equation which has the highest dimension. Thus the expression, for tan A, and so on. Or, gene sin 3A = 3 sin A 4 sin 3A (rad = 1) when radius is assumed = R, becomes 18. Supposing that in formulæ (c) A = B, and taking the superior sign, we shall have sin 2A 2 sin a cos a 2 sin a √(1) = 2A cos 24 = cos 24 sin A = 2 cox (Ain A) (D) - 1 by substituting 1 cos 2A for sin 2A, and 1 COS 2A. } • sin 2A for So, again, by supposing B.to be successively equal to 2A, 3A, 4A, &c. we shall find 19. If in the above expressions for the several multiples of the sines, we introduce for the cosines their values in terms of the sines, and in those for the multiples of the cosines the values of the sines in terms of the cosines, the following expressions will be obtained, in which each quantity is expressed in terms of its own kind. |