3

22(cos x)3 cos 3x + 3 cos x.

23 (cos x) = cos 4x + 4 cos 2x + 3.

2*(cos x)= cos 5x + 5 cos 3x + 10 cos a.

25 (cos x) = cos 6x + 6 cos 4x + 15 cos 2x + 10.

26 (cos x)= cos 7x+7 cos 5x + 21 cos 3x + 35 cos x.

2n-1 (cos x)"=cos nx+n cos (n−2)x+

n (n-1)
1. 2

cos (n−4)x+&c.

If n is odd, the number of terms is (n + 1), and the last

term is

1.3.5... nR

1.2.3... (n + 1)

2(n-1) cos ;

if n is even, the number of terms is n+1, and last term is

1.3.5...(n-1)
1.2.3... n

2n-1

The same observations may be applied to the expansion of (sin x)" in both these series n must be a positive integer. (L. 400-2; C. 503-10.)

SUMS OF TRIGONOMETRIC SERIES.

(35.) sin x + sin (x + a) + sin (x+2a) + &c. + sin(x+n-1.a)

sinna

sina

sin(x+n-1.a). B

sin x − sin (x + a) + sin (x + 2a) - &c. — sin (x+2r-1.a)

=

cos

sin ra a

cos(x+r-.a). ›

sin (x+a) + sin (x+2a)- &c. + sin (x + 2r-2.a)

cos x + cos(x + a) + cos (x + 2a) + &c. + cos (x + n − 1.a)

cos

cos (x + a) + cos (x + 2a) — &c. + cos (x + 2r-2.a)

sin x + 2 sin 2x + 3 sin 3x + &c. + n sin nx

cos x + 2 cos 2x + 3 cos 3x + &c. + n cos n x

(n + 1) cos nx · n cos (n + 1) x ·
4 (sin x)2

1

α

tan x + cot x + tan 2x + cot 2x + tan 4x + cot 4x + &c.

cosecx+cosec 2x + cosec 4x+ &c. + cosec 2"-1

(L. 403-16; C. 514-28; Hind, Trig. 298–318.)

- 1 co

cos 2x + cos 3x-cos 4a+ &c. = log. (2 cos x).°

Smm cos mx=

n

-1

(n + 1) cos nx ―n cos (n + 1) x

B Sm (tan 2m-1x + cot 2m-1,

4 (sin x)2

x) = 2 (cot x

= log. (2 cos x).

cos x + cos 2x + cos 3x+cos 4x + &c. - log, (2 sin x). (Lacr. Diff. Calc. Tome 2, 466.)

α

y=n sina-n2 sin 2x+n3 sin 3x-n1sin 4x + &c.

4

В

(36.) Resolution of trigonometrical quantities into factors.

cos =

α

&c.

B

= (1-4) (1-3) (1-1) &

2

(3π):)

2.2.4.4.6.6.8.8. &c.

2 1.3.3.5.5.7.7.9. &c.

γ

(L. 420-5: Lacr. Diff. Calc. Tome 3, Ch. vi.)

APPROXIMATE SOLUTION OF TRIANGLES.

(37.) Given A, B, c; A and B being small,

a+b-c=1ABc, nearly.

Given a, b, and the contained angle, π-0; 0 being small,