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lues of b and c for them in the preceding equation, and mul tiplying the whole by it will become

sin. A


sin. A sin. B. cos. c + sin. c. cos. B.

But, in every plane triangle, the sum of the three angles is two right angles; therefore, B and C are equal to the supplement of A: and, consequently, since an angle and its supplement have the same sine (cor. 1, pa. 3, vol. ii), we have sin. (B+c) = sin. B . cos. c + sin. c. cos. B.

10. If, in the last equation, c become subtractive, then would sin. c manifestly become subtractive also, while the cosine of c would not change its sign, since it would still continue to be estimated on the same radius in the same direction. Hence the preceding equation would become sin, (B-c) sin. B. cos. C sin. c

11. Let c' be the

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complement of c, and 4 be the quarter of the circumference: then will c′ = 40-c, sin. c'=cos. c, and cos. c' sin. c. But (art. 10), sin. (B — c') = sin. B. cos. c-sin. c' cos. B. Therefore, substituting for sin. c', cos. c', their values, there will result sin. (Bc') = sin. B . sin. c - Cos. B. cos. C. But because c = 10 c, we have sin. (B-C) sin. (B+C-40) = sin. [(B+c)−÷O] = sin. [O (B+c)]= cos. (B+c). Substituting this value of sin. (B-c') in the equation above, it becomes cos. (B+c) = COS. B. COS. C sin. B. sin. c.


12. In this latter equation, if c be made subtractive, sin. c will become sin. c, while cos. c will not change: consequently the equation will be transformed to the following, viz, cos. (B-c) = COS. B cos. c + sin. B. sin. c.

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If, instead of the angles B and C, the angles had been a and B; or, if A and B represented the arcs which measure those angles, the results would evidently be similar they may therefore be expressed generally by the two following equations, for the sines and cosines of the sums or differences of any two arcs or angles:

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13. We are now in a state to trace completely the mutations of the sines, cosines, &c, as they relate to arcs in the various parts of a circle; and thence to perceive that tables which apparently are included within a quadrant, are, in fact, applicable to the whole circle.

Imagine that the radius MC of the circle, in the marginal figure, coinciding at first with AC, turns about the point c (in the same manner as a rod would turn on a pivot), and thus










forming successively with AC all possible angles: the point м at its extremity passing over all the points of the circumference ABA'B'A, or describing the whole circle. Tracing this motion attentively, it will appear, that at the point A, where the arc is nothing, the sine is nothing also, while the cosine does not differ from the radius. As the radius MC recedes from AC, the sine PM keeps increasing, and the cosine CP decreasing, till the describing point м has passed over a quadrant, and arrived at B in that case, PM becomes equal to CB the radius, and the cosine CP vanishes. The point м continuing its motion beyond B, the sine P'M' will diminish, while the cosine cr', which now falls on the contrary side of the centre c will increase. In the figure, P'M' and CP' are respectively the sine and cosine of the arc a'm', or the sine and cosine of ABM', which is the supplement of A'M' to O, half the circumference: whence it follows that an obtuse angle (measured by an arc greater than a quadrant) has the same sine and cosine as its supplement; the cosine however, being reckoned subtractive or negative, because it is situated contrariwise with regard to the centre c.

When the describing point м has passed over O, or half the circumference, and has arrived at a', the sine P'M' vanishes, or becomes nothing, as at the point A, and the cosine is again equal to the radius of the circle. Here the angle ACM has attained its maximum limit; but the radius CM may still be supposed to continue its motion, and pass below the diameter AA'. The sine, which will then be "M", will consequently fall below the diameter, and will augment as M moves along the third quadrant, while on the contrary CP", the cosine, will diminish. In this quadrant too, both sine and cosine must be considered as negative; the former being on a contrary side of the diameter, the latter a contrary side of the centre, to what each was respectively in the first quadrant. At the point B', where the arc is three-fourths of the circumference, or O, the sine P"M" becomes equal to the radius CB, and the cosine CP" vanishes. Finally, in the fourth quadrant, from B' to A, the sine P"M"", always below AA', diminishes in its progress, while the cosine cr", which is then found on the same side of the centre as it was in the first quadrant, augments till it becomes equal to the radius CA. Hence, the sine in this quadrant is to be considered as pega


tive or subtractive, the cosine as positive. If the motion of M were continued through the circumference again, the circumstances would be exactly the same in the fifth quadrant as in the first, in the sixth as in the second, in the seventh as in the third, in the eighth as in the fourth: and the like would be the case in any subsequent revolutions.

14. If the mutations of the tangent be traced in like manner, it will be seen that its magnitude passes from nothing to infinity in the first quadrant; becomes negative, and decreases from infinity to nothing in the second; becomes positive again, and increases from nothing to infinity in the 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 40, in equa. v, it will become

cos. (B) = cos. 4O. cos. BF sin. 10. sin. B, sin. (O+B) = sin. O. cos. B sin. B. COS. 40. But sin. O = rad. = 1; and cos. 40 = 0:

so that the above equations will become

Cos. (B) = sin. B.

sin. (OB) = 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 O and less than 40 will be negative, but its sine positive. If в also be made = 0; then shall,

we have cos. O

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Suppose next, that in the equa. v, A0; then shall we obtain

cos, (B) =— COS. B.

sin. (O±3) = sin. B;

which indicates, that every arc comprised between 0 and 40, or that terminates in the third quadrant, will have its sine and its cosine both negative. In this case too, when B=0, or the arc terminates at the end of the third quadrant, we shall have cos. O = 0, sin. 40 = 1.

Lastly, the case remains to be considered in which A=0, or in which the arc terminates in the fourth quadrant. Here the primitive equations (V) give

cos. (B) = ± sin. B.
sin. (O+B)


so that in all arcs between 0 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 pre



ceding results to the algebraic expression for the tangent; viz, For it is hence manifest, that when the sine and 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 cotangent, viz, cot. = will produce exactly the same results.





The expressions for the secants and cosecants, viz, sec. = show, that the signs of the secants are the 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 positive, the latter negative.



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.

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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 R, 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 Ras shall make it of the same dimension, as the term in the equation which has the highest dimension. Thus, the expression for a triple arc

sin. 3A 3 sin. A

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4sin3. A (radius = 1) becomes when radius is assumed = R,

R2 sin. 3A = R2 3 sin. A - 4 sin3. A

or sin. 3A =

3R2 sin. A-4 siñ3. A


Hence then, if consistently with this precept, R be placed for a denominator of the second member of each equation v (art. 12), and if a be supposed equal to B, we shall have


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sin. A cos. A + sin. A COS.A

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2 sin. A. cos. A


And, in like manner, by supposing B to become successively equal to 2A, 3A, 4A, &c, there will arise

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And, by similar processes, the second of the equations just referred to, namely, that for cos. (A + B), will give successively,

cosa. A

sin2. A

COS. 2A =

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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 cosine.


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