COR. 1. By transposition,

re-r.cos. 2A, or r(r-cos. 2A)=2 sin.2 A (see p. 11.).

Now, r - cos. 2 A = ver, sin. 2 A, consequently,

r.(ver. sin. 2 A) = 2 sin. A (see p. 11.)

which equality, thrown into a proportion, is

ver. sin. 2 A : sin. A :: sin. A :

112

or, expressed in general terms, announces, that the sine of an angle is a mean proportional between the versed sine of twice the same angle and the semi-radius.

Again, r2 + r. cos. 2 A = 2 cos.2 A,
orr(r + cos. 2 A) = 2 cos. A,

orr (ver. sin. supp2. of 2 A) = 2 cos. A ;

or, (if we call the ver. sin. of the supplement of an arc, the suversed sine) r. suversin. (2 A) = 2 cos. A, which equality, like the preceding, may be expressed under the form of a proportion, or in general terms.

COR. 2. Hence the sine of 30° =, radius = 1,

For, sin. 60° = sin. (2.30°) = 2 sin. 30°. cos. 30°, [Prob. 5.],"

but, cos. 30°=sin (90° - 30°) = sin. 60°;

... sin. 60° = 2 sin. 30°. sin. 60°;

or sin. 30° , and consequently sin. 60°, or cos. 30°=√3.

=

COR. 3. Since, radius being 1,

sin. A + cos. A = 1,

and, (p. 12.) 2 sin. A. cos. A = sin. 2 A,

we have, by the solution of a quadratic equation such as

x2 + y2 = a,

2

in which expressions the upper or lower sign that affects the second term is to be used, accordingly as sin. A is greater or less than cos. A, that is, in arcs not exceeding a quadrant, accordingly as A is greater or less than half a quadrant.

These two formulæ are useful either to compute arithmetically the sines, cosines, &c. of arcs, or to examine their accuracy when computed by other formule; and, performing the latter office, they are called Formula of Verification. It is easy to perceive their use in computing sines, cosines, &c.; since, if we take sin. 2 A, a known quantity, for instance, the sin. 30° which equals, we may, by successive substitutions, regularly deduce the sines of 15°, 7° 30′, &c. Thus,

sin. 15° = √(1+)-√(1-1) =.258819,

sin. 7° 30′ = √(1.258819) - √(.741181) = .1305262, sin. 3° 45′ =, &c.

PROBLEM 6. It is required to express the sine and cosine of 3 times an arc, 4 times an arc, &c. in terms of the sine and cosine of the simple arc.

If we substitute in the formula for cos. (A+B) (p. 29.) 2 instead of B, we have

cos. (A + 2 A) = cos. 2 A. cos. A

but, by the preceding Problem,

... cos. 3

sin. 2 A. sin. A;

sin. 2 A = 2 sin. A cos. A, and cos. 2 A = 2 cos. A

1;

= (2.cos. A
2 cos. A. sin.2 A
1) cos. A
= 2. Cos.3 A cos. A
cos.2
2 cos. A (1
= 4 cos.3 A - 3 cos. A, when the radius = 1,
4 cos.3 A
3. cos. A, when the radius = r.

This form, if we substitute therein, instead of the arc 3 A, gives us

cos. (-3.4) = - sin. 34, cos. (3-4) = sin. 4;

π

2

consequently,

sin. 3 A 3 sin. A

= 3 r2. sin. A

A

4. sin.3 A, when the radius = 1,

4. sin.3 A

2

A;

when the radius = r.

By a similar method may the cos. (4A) = cos. (3 A + A) or= Cos. (2 A + 2A), and the cos. (5 A) = cos. (4 A + A) or = cos. (3 A+ 2 A), &c. be deduced. But, the successive formation of the cosines and sines of multiple arcs may, most easily, be effected after the following manner:

By the form [d], page 31,

cos. (A+B) + cos. (A - B) = 2 cos. A.cos. B...... A > B, or cos. (B+A) + cos. (B - A) = 2 cos. B. cos. A......B> A.

Let B = n A, then transposing

cos. (n + 1) A = 2.cos. nA.cos. A cos. (n - 1) A, and hence from cos. (n - 1) A, and cos. nA, may be assigned cos. (n + 1) A: for instance,

if n = 1, cos. (n - 1) A = cos. 0 = 1;

:. cos. 2A = 2 cos. A - 1........ .......[].

If n = 2,

cos. 3A-2.cos. 2 A.cos. A-cos. A=4.cos. A - 3 cos. A [11].

If n = 3,

cos. 4 A = 2 cos. 3A.cos. A - cos. 2 A

= 2(4 cos. A-3 cos. A) cos. A - (2 cos. A -F)
= 8 cos. A
8 cos. A + 1..........[],

And, by a like process, if n = 4,
cos. 5 A =

16 cos. A + 2 cos. A - (4 cos. A - 3 cos. A)

= 16 cos. A-20 cos. A + 5 cos. A...[c],

or, 2 cos. 5 A = (2 cos. A) - 5 (2 cos. A) + cos. A.

Similarly, if [n = 5],

cos. 6 A = 32 cos.6 A - 40 cos.* A + 10 cos. A

- (8 cos. A 8 cos. A + 1)

= 32 cos.6 A-48 cos. A + 18 cos.2 A-1.... [c],

The formula for the sines of multiple arcs may be deduced from those of the cosines, and, on the same principle as that which has been already used in deducing sin. 3 A; by substituting, in the form for cos. 5 A,

π

A

5 π
2

5 A instead of 5 A, we have

cos.-54) =

5

2

T

A

3

16 (cos.-4) -20(cos. -1) + 5. cos. (-1),

2

+5.cos.

5.A)=

cos. (-A) = sin. 4;

A;

sin. 5 A,

consequently, sin. 5 A = 16 sin.5 A - 20 sin. A + 5 sin. A.

Or, the sines of multiple arcs may be successively deduced as the cosines have, on the same principle, and by like formulæ; thus, by the form [a], p. 31.

sin. (B+A) + sin. (B-A) = 2 sin. B. cos. A.

Let B = n A, then, transposing

sin. (n + 1) A = 2 sin. n A.Cos. A - sin. (n-1) A, hence, if n = 2,

sin. 3 A=2.sin. 2 A. cos. A – sin. A, (sin. 2 A = 2 sin. A. cos. A)=4 sin. A. cos. A-sin. A..[5"] =3 sin. A-4 sin. A...... [s].

Similarly, if n = 3,

sin. 4 A 2 cos. A (3 sin. A

1

4 sin.3 A) - sin. 2 A,

= 6 sin. A.cos. A-8 cos. A. sin. A - 2 sin. A.cos. A

= 5 sin. A

(3 sin. A

4 sin.3 A) 8 sin. A + 16 sin.5 A -- (3 sin. A

[5"],

4 sin.3 A) 20 sin.3 A + 16 sin.5 A or, 2 sin. 5 A = 5.2 sin. A - 5. (2 sin. A)3 + (2 sin. A)5.

..

..............

The general expression for sin. m A [m odd] is

sin. m A =

m.sin. Am. (m2 -1) sin.3 A + m (m2-1)(m2 -9) sin.5 A- &c.

:

2.3

and [m even] is

2.3.4.5

:

:

The sine and cosine of the multiple arc (mA) have been ex

* For the general demonstration of these forms, see the Appendix..