what parts of the circle each of them increases or decreases. But this does not determine their exact values, except at the extremities of the several quadrants. In the analytical investigations which are carried on by means of these lines, it is necessary to calculate the changes produced in them, by a given increase or diminution of the arcs to which they belong. In this there would be no difficulty, if the sines, tangents, &c. were proportioned to their arcs. But this is far from being the case. If an arc is doubled, its sine is not exactly doubled. Neither is its tangent or secant. We have to inquire, then, in what manner, the sine, tangent, &c. of one arc may be obtained, from those of other arcs already known.

The problem on which almost the whole of this branch of analysis depends, consists in deriving, from the sines and cosines of two given arcs, expressions for the sine and cosine of their sum and difference. For, by addition and subtraction, a few arcs may be so combined and varied, as to produce others of almost every dimension. And the expres sions for the tangents and secants may be deduced from those of the sines and cosines.

Expressions for the SINE and COSINE of the SUM and DIFFERENCE of arcs.

207. Let a=AH, the greater of the given arcs,
And b=HL=HD, the less. (Fig. 37.)

Then a+b=AH+HL=AL, the sum of the two arcs,
And a-b-AH-HD=AD, their difference.

Draw the chord DL, and the radius CH, which may be represented by R. As DH is, by construction, equal to HL; DQ is equal to QL, and therefore DL is perpendicular to CH. (Enc. 3. 3.) Draw DO, HN, QP, and LM, each perpendicular to AC; and DS and QB parallel to AC:

From the definitions of the sine and cosine, (Arts. 82, 9.) it is evident, that

The triangle CHN is obviously similar to CQP; and it is also similar to BLQ, because the sides of the one are perpendicular to those of the other, each to each. We have, then,

1. CH: CQ::HN: QP, that is, R: cos b::sin a: QP,

2. CH: QL::CN: BL,

3. CH: CQ::CN: CP,

4. CH: QL::HN: QB,

R sin b::cosa: BL,
R: cosb::cos a: CP,
R sin b: sin a : QB.

* In these formulæ, the sign of multiplication is omitted; sin a cos b being put for sin aX cos b, that is, the product of the sine of a into the cosine of b.

But it will be seen, from the figure, that
QP+BL=BM+BL=LM=sin (a+b)
QP-BL=QP-QS-DO=sin (a-b)

CP-QB CP-PM-CM=cos(a+b)
CP+QB CP+SD=CO=cos (a-b)

208. If then, for the first member of each of the four equations above, we substitute its value, we shall have,

III. cos(a+b)=cos a cos b—sin a sin b

IV. cos(a-b)=C

R

cos a cos b+sin a sin b

R

That is, the product of radius and the sine of the sum of two arcs, is equal to the product of the sine of the first arc into the cosine of the second + the product of the sine of the second into the cosine of the first.

The product of radius and the sine of the difference of two arcs, is equal to the product of the sine of the first arc into the cosine of the second the product of the sine of the second into the cosine of the first.

The product of radius and the cosine of the sum of two arcs, is equal to the product of the cosines of the arcs the product of their sines.

The product of radius and the cosine of the difference of two arcs, is equal to the product of the cosines of the arcs + the product of their sines.

These four equations may be considered as fundamental propositions, in what is called the Arithmetic of Sines and Cosines, or Trigonometrical Analysis.

Expressions for the sine and cosine of a DOUBLE arc.

209. When the sine and cosine of any arc are given, it is easy to derive from the equations in the preceding article, expressions for the sine and cosine of double that arc. As the two arcs a and b may be of any dimensions, they may be supposed to be equal. Substituting, then, a for its equal b, the first and the third of the four preceding equations will be

come,

R sin (a+a)=sin a cos a+sin a cos a
R cos (a+a)=cosa cos a-sin a sin a

That is, writing sina for the square of the sine of a, and cos'a for the square of the cosine of a,

I. R sin 2a=2sin a cos a
II. R cos 2a=cos2a-sin' a.

Expressions for the sine and cosine of HALF a given arc.

210. The arc in the preceding equations, not being necessarily limited to any particular value, may be half a, as well as a. Substituting then a for a, we have,

Putting the sum of the squares of the sine and cosine equal to the square of radius, (Art. 94.) and inverting the members of the last equation,

cos2a+sin2a=R2
cos2a-sin2 a=R cos a

If we subtract one of these from the other, the terms containing cosa will disappear; and if we add them, the terms containing sin2 a will disappear: therefore,

Dividing by '2, and extracting the root of both sides,

I. sina=√R2 – R×cos a

Expressions for the sines and cosines of MULTIPLE arcs.

211. In the same manner, as expressions for the sine and cosine of a double arc, are derived from the equations in art. 208; expressions for the sines and cosines of other multiple arcs may be obtained, by substituting successively 2a, 3a, &c. for b, or for b and a both. Thus,

I.

II.

R sin 3a=R sin(a+2a)=sin a cos 2a+sin 2a cos a
R sin 4a=R sin(a+3a)=sin a cos 3a+sin 3a cos a
R sin 5a-R sin(a+4a)=sin a cos 4a+sin 4a cos a
&c.

R cos 3a=R cos(a+2a)=cos a cos 2a - sin a sin 2a
R cos 4a=R cos(a+3a)=cos a cos 3a-sin a sin 3a
R cos 5a=R cos(a+4a)=cos a cos 4a-sin a sin 4a
&c.

Expressions for the PRODUCTS of sines and cosines.

212. Expressions for the products of sines and cosines may be obtained, by adding and subtracting the four equations in art. 208, viz.

R sin(a+b)=sin a cos b+sin b cos a
R sin(a-b)=sin a cos b- sin b cos a
Rcos(a+b)=cosa cosb-sin a sin b
Rcos(a-b)=cosa cosb+sin a sin b
Adding the first and second,
R sin(a+b)+R sin(a-b)=2 sin a cos b
Subtracting the second from the first,
R sin(a+b)-R sin(a-b)=2sin b cos a
Adding the third and fourth,

R cos(a-b)+R cos(a+b)=2cos a cos b
Subtracting the third from the fourth,
R cos(a-b) -R cos(a+b)=2sin a sin b