Page 695 - Handbook Of Integral Equations

Supplement 1

Elementary Functions

and Their Properties

Throughout Supplement 1 it is assumed that n is a positive integer, unless otherwise specified.

1.1. Trigonometric Functions

Simplest relations
2
2
sin x + cos x = 1, sin(–x)=– sin x, cos(–x) = cos x,
sin x cos x 2 1 2 1
tan x = , cot x = , 1 + tan x = , 1 + cot x = ,
cos x sin x cos x sin x
2
2
tan x cot x = 1, tan(–x) = – tan x, cot(–x) = – cot x.
Relations between trigonometric functions of single argument
√ tan x 1
2
sin x = ± 1 – cos x = ± √ = ± √ ,
2
2
1 + tan x 1 + cot x
√ 1 cot x
2
cos x = ± 1 – sin x = ± √ = ± √ ,
2
2
1 + tan x 1 + cot x
√
2
sin x 1 – cos x 1
tan x = ± √ = ± = ,
2
1 – sin x cos x cot x
√
2
1 – sin x cos x 1
cot x = ± = ± √ = .
sin x 1 – cos x tan x
2
Reduction formulas
n
n
sin(x ± nπ) = (–1) sin x, cos(x ± nπ) = (–1) cos x,
2n +1 n 2n +1 n

sin x ± π = ±(–1) cos x, cos x ± π = ∓(–1) sin x,
2 2
tan(x ± nπ) = tan x, cot(x ± nπ) = cot x,
2n +1 2n +1

tan x ± π = – cot x, cot x ± π = – tan x,
2 2
√ √
π 2 π 2

sin x ± = (sin x ± cos x), cos x ± = (cos x ∓ sin x),
4 2 4 2
π tan x ± 1 π cot x ∓ 1

tan x ± = , cot x ± = .
4 1 ∓ tan x 4 1 ± cot x
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 679