Metamath Proof Explorer


Theorem sinper

Description: The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008) (Revised by Mario Carneiro, 10-May-2014)

Ref Expression
Assertion sinper
|- ( ( A e. CC /\ K e. ZZ ) -> ( sin ` ( A + ( K x. ( 2 x. _pi ) ) ) ) = ( sin ` A ) )

Proof

Step Hyp Ref Expression
1 sinval
 |-  ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
2 sinval
 |-  ( ( A + ( K x. ( 2 x. _pi ) ) ) e. CC -> ( sin ` ( A + ( K x. ( 2 x. _pi ) ) ) ) = ( ( ( exp ` ( _i x. ( A + ( K x. ( 2 x. _pi ) ) ) ) ) - ( exp ` ( -u _i x. ( A + ( K x. ( 2 x. _pi ) ) ) ) ) ) / ( 2 x. _i ) ) )
3 1 2 sinperlem
 |-  ( ( A e. CC /\ K e. ZZ ) -> ( sin ` ( A + ( K x. ( 2 x. _pi ) ) ) ) = ( sin ` A ) )