Basic mathematics Using trigonometric identities Rele cosb) =cos acos=1/2[cos(a+b)+cos(a-b)] Im(e cosb )=sinacosb=1 2 [ sin(a+b)-sin(a-b) Using these relationships we can derive the output obtained by multiplying be cos and sin versions of the outgoing signal are V(tcos2ft=1 2GE cos2rkx+1/2cos(2 k x.+4nft) V(tsin2nft=1 2GEo sin2k x +1 2sin(2 rk. x +4ft) 02/25/02 12.540Lec0602/25/02 12.540 Lec 06 15 Basic Mathematics • Using trigonometric identities: • Using these relationships we can derive the output obtained by multiplying be cos and sin versions of the outgoing signal are Re( e −ia cos b ) = cos acos b =1 2 [cos( a + b ) + cos( a − b ) ] Im( e −ia cos b ) = sin acos b =1 2[ ] sin( a + b ) − sin( a − b ) V ( t )cos 2 πft =1 2GE 0 cos 2 πk.x 0 +1 2cos(2 πk.x 0 + 4 πft ) V ( t )sin 2 πft =1 2GE 0 sin 2 πk.x 0 +1 2sin(2 πk.x 0 + 4 πft )