2 (cos x) = cos 2x + 1.
2°(cos x)3 =cos 3x + 3 cos X.
23(cos x)4 = cos 4x + 4 cos 2x + 3.
2(cos x)" =cos 5x + 5 cos 3x + 10 cos a.
2(cos x)® = cos 6x + 6 cos 4x + 15 cos 2x + 10.
2(cos x)?=
=cos 7x + 7 cos 5x + 21 cos 3x + 35 cos al.

n(n-1) 2n-1 (cos x)"= =cos n& +n cos (n-2)x+ cos (n—4) +&c.

1. 2 If n is odd, the number of terms is į (n + 1), and the last

1.3.5... term is

1.2.3... (n+1) if n is even, the number of terms is n+1, and last term is

1.3.5...(n-1)

1.2.3... in 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.)

n

2}(n-1) cos x;

..2n-1

SUMS OF TRIGONOMETRIC SERIES. (35.) sin x + sin (x + a) + sin (x + 2a) + &c. + sin (x + 1-1.a) sinna

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

sin sa

sin a

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

sin ra

cosa

cos (x+r-.a).

sin na So, sin (x+m-1.a)= sin (x + {n-1.a).

sin ra

sin (x +r-5.a). y Sm(-1)"- 1 sin (« + m-1.a)=

cosa

sin

a

2r

S

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

sin (x+p-1.a). cos ka

a

a

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

cos (x + 1ñ– 1.a). B

sin na sin La

COS NO

cos (x + a) + cos (x + 2a) - &c. - cos (x + 2r-1.a)

COS X

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

cos (0.+ 99 1.a).
ha

COS

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

(n + 1) sin na n sin (n + 1) x

4 (sin x)

2r-1

cos (r – })a sin (x+r-1.a).

a

n

Sm (-1)" – 1 sin (x + m - 1.a) =

cosa B Socos (x + m - 1.a) =

sinna

cos (x + {n-1.a)

cosa v sm (-1)" – 1 cos (x +m-1.a) sin (x +-.a). .

cosa

cos (r->) a Sm(-1)M–1 cos (x +m-1.a)=

cos (x+r-1.a). cos

2r

sin ra

2r-1

a

n

(n + 1) sin na n sin (n + 1) a Sym sin mx=

4 (sin ? x)

COS &C + 2 cos 2 x + 3 cos 3x + &c. + n cos n x (n + 1) cos nxn cos (n + 1) x — -1

4 (sin ? x)?

a

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

+ tan 21 -12 + cot 2n = 2 (cot X – cot 2" x).

1

B

=

cosec x + cosec 2x + cosec 4x + &c. + cosec 21 – 1 2

=cot. x-cot 2n-18.

sin x – { sin 2x + } sin 3x – 1 sin 4x + &c. = 1x !

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

cos 2x + { cos 3x - 1 cos 4x + &c. = logo (2 cos 1 x).

n

a

(n + 1) cos n x — n cos (n + 1)x -1. Smm cos mx=

4 (sin x)? B Sm (tan 2m – 1x + cot 2m –x)=2 (cot & - cot 2" x).

n

a

If tany=

cos æ+ cos2x+cos3x+cos 4x + &c. =-- log. (2 sin ? x).

(Lacr. Diff. Calc. Tome 2, 466.) n sin a

then

1 +n cos a y=n sin x - , n* sin 2x + į no sin 3x- n* sin 4x + &c.

(C. 513.) If tan y=n tan ,

then 1-n sin 2x

2 sin 4x

3 sin 6 x y=

+
1+n
1
1+n

+n 3
(Maddy, Astron. 62.)

n

+&c.

Y

sin a

+

n

n

2

COS X

.

2 1

{ {}+ift +{} + } cos w}*.

2n {1+}{t+}{1+ +}{1+ £cosæ}*.

2n

(Hind, Trig. 81.) (36.) Resolution of trigonometrical quantities into factors.

37 57 sin (2n-1) 1 sin

.sin 2n

2n

2n

T

sin 2n

&

Fy
Fy) 1.

&c.

4 (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=} ABc, nearly. Given a, b, and the contained angle, -0; being small,

ab02
c=a+b-

nearly;
2(a+b)
αθ ab(a-6) 03
+

(a + b)? 7 · nearly. (Leg. 97-9.)

A=

a+b