Multiple-Angle Formulas

Formulas expressing trigonometric functions of an angle  nx  in terms of functions of an angle x.

 
sin ( 2 x ) = 2 sin x  cos x 
sin ( 3 x ) = 3 sin x  - 4 sin3 x 
sin ( 4 x ) = cos x (4 sin x - 8 sin3 x)
sin ( 5 x ) = 5 sin x - 20 sin3 x  + 16 sin5 x 
cos ( 2 x ) = cos2 x -  sin2 x 
sin ( 3 x ) = cos3 x  - 3 (cos x) (sin2 x )
cos ( 4 x ) = cos4 x  - 6 (cos2 x) (sin2 x ) + sin4 x
cos ( 5 x ) = cos5 x  - 10 (cos3 x) (sin2 x ) + 5 (cos x) (sin4 x)

 
sin ( n x ) = 2 sin [( n - 1 )x ] cos x - sin [( n  - 2) x]
cos ( n x ) = 2 cos [( n - 1 )x ] cos x - cos [( n  - 2) x]
tan ( n x ) ={ tan [( n - 1 ) x] + tan x } / {1 - tan [( n -1 ) x ] tan x }

Trigonometric Power Formulas

sin2 x = 1/ 2 [ 1 - cos ( 2x )]
sin3 x = 1/4 [3 sin x - sin ( 3x )]
sin4 x = 1/8 [3 - 4 cos (2x) + cos ( 4x )]
cos2 x = 1/2 [1 + cos ( 2x )]
cos3 x = 1/4 [3 cos x + cos ( 3x )]
cos4 x = 1/8 [3 + 4 cos (2x) + cos ( 4x )]