This is the fundamental form, from which almost all other Trí gonometrical forms may be deduced.

COR. J.

cos. (A + B) = (p. 6.) sin. ( - (A + B)) =

obtained by direct investigation, but discovered to be true by observation on the values of sin. A, cos. A, sin. B, cos. B, and by previous knowledge of those values. There is no difficulty in directly investigating it, by means of a construction (see Thomas Simpson's Trig. p. 54.) An able critic, however, (see Edin. Review, No. XXXIII. p. 128.) suggests, that this, and similar formulæ, may be naturally and simply deduced by means of a Lemma of Ptolemy in his μεγαλη συνταξις, and which in Simpson's edition of Euclid is the Prop. D of the Sixth Book. Thus, according to that proposition,

AB × CD + BC × AD = AC x BD.

Let the arc AD = DC = 2B, and the arc BAD = 2 A;

then,

(AB + BC) AD = AC × BD, or, (see p. 10.)

{2.sin. (A-B)+2 sin. (A+B) } 2 sin. B=2sin. 2B x 2 sin. A (a).

X

{sin. (B)+sin. (+B)} sin. B = sin. 2 Bx sin.

π

π

2

π

but, (pp. 6, 13.) sin. (-B,) and sin. (B)=cos. B and sin. =1;

2

2. cos. Bx sin. B = sin. 2 B, we have (the same as in p. 12.) and substituting this in the form (a) 1. 18. we have

p. 15.) sin. ( + (A+B));

but, sin. (+ (A + B))=sin. {(+1) + B} .

(b).

sin. (A - B) + sin. (A + B) = 2 sin. A. cos. B Since in this formula A and B may be any arcs provided that A > B,

π

π

substitute + A instead of A, and 2 + B instead of B, then

sin. (A - B) - sin. (A + B) = - 2 cos. A. sin. B (c). Hence, by adding and subtracting (b) and (c), there will result sin. (A - B) = sin. A. cos. B - cos. A. sin. B,

and sin. (A+B) = sin. A.cos. B + cos. A.sin. B,

and the deduction of other forms from these fundamental ones may be conducted as it is in the text.

Mr. Cresswell, in his Treatise on Spherics, has more simply demonstrated the latter of these fundamental formulæ: thus

+A

= sin. (+4) cos. B+ cos. (+4). sin. B,

which is derived from the form [1], by substituting, instead of 4, (+4).

Hence, cos. (A + B) = cos. A. cos. B - sin. A. sin. B....[2]. COR. 2. By page 6,

sin. (A - B)=cos. ( - (AB)) = cos. {(-A)+B}.

But, by the formula [2] that has been just established,

cos. (-4) cos. B-sin. (-4) sin. B

A

A

Ds = BD.cos. BDA = 2 sin. B cos. A;

... sin. (A + B) = sin. A cos. B + cos. A. sin. B.

Again, by page 6,

cos. (A-B) = sin. ( - (A - B)) = sin. {(-A) +B}.

But, by the formula [1],

A

sin. {(-4)+B}=sin. (-4) cos. B+cos. (-4) sin. B

Hence, therefore,

COS.

-A

= cos. A cos. B + sin. A. sin. B (by p. 6.)

* cos. (A - B) = cos. A. cos. B + sin. A. sin. B........[4].

COR. 3. Add together the forms (1) and (3,] and there results sin. (A+B) + sin. (A - B) = 2.sin. A. cos. B......[a]. Subtract [3] from [1], and

sin. (A + B) - sin. (A - B) = 2.cos. A.sin. B......[b].

Multiply [1] and [3],

and, the right-hand side of the equation is =

Hence,

sin. Ax cos. B - cos. A x sin. B =

sin. A (1 - sin.2 B) - (1 - sin. A) sin. B =

sin. (A + B) x sin. (A - B) = sin. A

Add [2] and [4], and

Subtract [2] from [4], and

cos. (A + B) = 2 sin. A. sin. B......[e].

If we substitute in the preceding formule [a], [b], [c], &c.

the quantity (n + 1) B instead of A, we shall have

sin. (n+2) B + sin. n B = 2 sin. (n + 1) B cos. B,

sin. (n+2) B - sin. n B = 2 cos. (n + 1) B sin. B,

sin. (n+2) B x sin. n B = sin. (n + 1) B - sin.2 B,
cos. n B + cos. (n+2) B = 2 cos. (n+1) B.cos. B,
cos. n B
cos. (n+2) B = 2 sin. (n + 1) B.sin. B.

COR. 4. Some of the preceding forms may be differently expressed, for

S D

since A=A+B+A=B = + making

and

2

2

=

2 2 S = A + B,

S

B=A+BA-B-DD=A-B,

2

we have from [a],

2

2 2

sin. S + sin. D = 2 sin. (S+D).cos. (S-D),

and from [b],

2

sin. S - sin. D = 2 cos. (S+D). sin. (SD),

2

2

or, since S and D are any arcs subject to this condition alone, namely, that S > D, and since, in a series of formulæ, it is convenient to use the same characters, instead of S and D we may use A and B, and then,

2

sin. A + sin. B = 2 sin. (A+B).cos. (A - B) ...... [5], sin. A - sin. B = 2 cos. (A+B). sin. (A-B)......[6].

By a similar process we may transform [d] and [e] into these,