sin. B=2✓✓/}S({s—a) (¦s—b) (1s – c

(III.)

sin. c=(-a) (1s-b) (1s-c

2 ab

These equations are moderately well suited for computation in their latter form; they are also perfectly symmetrical: and as indeed the quantities under the radical are identical, and are constituted of known terms, they may be represented by the same character; suppose K: then shall we have

sin. A

2K bc

2K

sin. B =

aç

2K

sin. c

ab

(iii.)

Hence we may immediately deduce a very important theorem: for, the first of these equations, divided by the second

gives

sin. A a
sin. B b

and the first divided by the third gives

sin. A

sin. c

Or, in words, the sides of plane triangles are proportional to of their opposite angles. (See th. 1, Trig. vol. i).

9. Before the remainder of the theorems, necessary in the solution of plane triangles, are investigated, the fundamental proposition in the theory of sines, &c. must be deduced, and the method explained by which Tables of these quantities, confined within the limits of the quadrant, are made to extend to the whole circle, or to any number of quadrants whatever. In order to this, expressions must be first obtained for the sines, cosine, &c. of the sums and differences of any two arcs or angles. Now, it has been found (1) that ab. cos. ctc. cos. B. And the equations (IV) give b=a.

in the preceding equation, and multiplying the whole by

it will become

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

B

sin. A

a

But, in every plane triangle, the sum of the three angles is two right angles; therefore в and c are equal to the supplement of a: and, consequently, since an angle and its supplement have the same sine (cor. 1, pa. 378, vol, i), we have sin. (B+c) sin. B. cos. c+ sin. c. cos, B..

10. If,

1

10. If, in the last equation, c become subtractive, ther 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

COS. B.

11. Let be the complement of c, and 10 he the quarter of the circumference: then will c'=40-c. sin. c'= cos. c, and cos. c' sin. c. But (art. 10), sin. 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. (B-c') = sin. B sin. c.COS. B. COS. c. but because c' cos. = 0 C, we have sin. (B-C) sin. (B+C-10)= sin. [(B+c) - 10]=sin. [-(B+c)]=cos. (B+c). Substituting this value of sin. (B-) 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. COS.

A

If, instead of the angles B and C, the angles had been ▲ and. B; or, if ▲ and в 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 : sin. (A+B)

COS. (A+B)

sin.

sin. A . cos. B. ± sin. B. cos. A.
Cos. A cos. B. sin. A. sin. B.

}

(V.)

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 ċ (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

B

B'

from

༦, *

M

from the radius. As the radius Mc recedes from Ac, the sine
Pм keeps increasing, and the cosine cr 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 be-
yond 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, PM' 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 AM to 10 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, be-
cause it is situated contrariwise with regard to the centre c.
When the describing point м has passed over 10, 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 r"m", will consequently
fall below the diameter, and will augment as м moves along
the third quadrant, while on the contrary cr" 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
20, the sine P"M" becomes equal to the radius CB, and the
cosine CP vanishes. Finally, in the fourth quadrant, from вʻ
to A, the sine "M", always below AA', diminishes in its pro-
gress, while the cosine CP", which is then found on the same
side of the centre as it was in the first quadrant, auguments
till it becomes equal to the radius CA. Hence, the sine in this
quadrant is to be considered as negative or subtractive, the
cosine as positive. If the motion of м 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 quad

rant;

rant; 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

A

COS. COS. B sin. 1. sin. B, sin. (10B) sin.. cos. B± sin. B cos. 10. But sin. 1 = rad. =1 1; and cos. † O=0:

so that the above equations will become

COS. (B)

sin. (B)

I sin. B.

COS. B.

will be negative,

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 40 and less than
but its sine positive. If в also be made
we have cos. O 1; sin. ¦ 0 = 0.
Suppose next, that in the equa. v, A

obtain.

COS. (+B=

O

O; then shall

; then shall we

COS. B.

sin. (B = ± sin. B;

which indicates, that every arc comprised between

and

when

O, or that terminates in the third quadrant, will have its sine and its cosine both negative. In this case too, B=1○, or the arc terminates at the end of the third quadrant, we shall have cos. = 0, sin. 30

}

4

- 1.

Lastly the case remains to be considered in which a=30 or in which the arc terminates in the fourth quadrant. Here the primitive equations (V) give

so that in all arcs between 3
tive and the sines negative.

tan.

sin.

COS.

± sin. B.

COS. B';

and O, the cosines are posi

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. 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. will produce exactly the same results.

cot.

COS.
sin,

The

The expressions for the secants and cosecants, viz. sec. =

COS.

cosec.

1 sin.

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 sin. =co (infinity). Of these, however, the former will be reckoned positive, the latter negative..

COS.

17. The magnitudes of the cotangents, secants, and cosecants may be traced in like manner, and the results of the and tabulated as

13th, 14th, and 15th articles, recapitulated results of the

below.

Sin. 0
Tan. 0

88

Cosec.com

We have been thus particular in tracing the mutations, both with regard to the 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.EP

generality, give

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 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 terms in the equation which has the highest dimension. Thus the expression for a triple arc.