Page 697 - Handbook Of Integral Equations

Trigonometric functions of multiple arguments

2
2
cos 2x = 2 cos x – 1=1 – 2 sin x, sin 2x = 2 sin x cos x,
3
3
cos 3x = –3 cos x + 4 cos x, sin 3x = 3 sin x – 4 sin x,
3
2
4
cos 4x =1 – 8 cos x + 8 cos x, sin 4x = 4 cos x (sin x – 2 sin x),
5
3
5
3
cos 5x = 5 cos x – 20 cos x + 16 cos x, sin 5x = 5 sin x – 20 sin x + 16 sin x,
n 2 2 2 2
k n (n – 1) ... [n – (k – 1) ] k 2k
cos(2nx)=1 + (–1) 4 sin x,
(2k)!
k=1
n
2 2 2 2 2
k [(2n+1) –1][(2n+1) –3 ] ... [(2n+1) –(2k–1) ] 2k
cos[(2n+1)x] = cos x 1+ (–1) sin x ,
(2k)!
k=1
n
2 2 2 2 2
k (n – 1)(n – 2 ) ... (n – k ) 2k–1
sin(2nx)=2n cos x sin x + (–4) sin x ,
(2k – 1)!
k=1
n
2 2 2 2 2
k [(2n+1) –1][(2n+1) –3 ] ... [(2n+1) –(2k–1) ] 2k+1
sin[(2n+1)x]=(2n+1) sin x+ (–1) sin x ,
(2k+1)!
k=1
3
3
2 tan x 3 tan x – tan x 4 tan x – 4 tan x
tan 2x = , tan 3x = , tan 4x = .
4
2
2
2
1 – tan x 1 – 3 tan x 1 – 6 tan x + tan x
Trigonometric functions of half argument
x 1 – cos x 2 x 1 + cos x
2
sin = , cos = ,
2 2 2 2
x sin x 1 – cos x x sin x 1 + cos x
tan = = , cot = = ,
2 1 + cos x sin x 2 1 – cos x sin x
2 tan x 2 1 – tan 2 x 2 2 tan x 2
sin x = 2 x , cos x = 2 x , tan x = 2 x .
1 + tan 1 + tan 1 – tan
2 2 2
Euler and de Moivre formulas. Relationship with hyperbolic functions
y
n
2
e y+ix = e (cos x + i sin x), (cos x + i sin x) = cos(nx)+ i sin(nx), i = –1,
sin(ix)= i sinh x, cos(ix) = cosh x, tan(ix)= i tanh x, cot(ix)= –i coth x.
Differentiation formulas
d sin x d cos x d tan x 1 d cot x 1
= cos x, = – sin x, = , = – .
2
2
dx dx dx cos x dx sin x
Expansion into power series
x 2 x 4 x 6
cos x =1 – + – + ··· (|x| < ∞),
2! 4! 6!
x 3 x 5 x 7
sin x = x – + – + ··· (|x| < ∞),
3! 5! 7!
x 3 2x 5 17x 7
tan x = x + + + + ··· (|x| < π/2),
3 15 315
1 x x 3 2x 5
cot x = – – – – ··· (|x| < π).
x 3 45 945
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 681