Signals &systems 10EE52  sin( cos(kw0 0 gw0 t) t)  0 for  2f w0 1  f T and and are integers. k g Solution Let T cos(kw0t)dt (15) C  gw0 t) 0 Recall that sin( sin() cos() sin() cos() (16)  )   Hence, T w0  sin(kw0 cos(gw0  k t t) t)dt (17) C 0 T T sin(g )w0tdt  k    sin(kw0 cos(gw0  (18) t) t)dt 0 0 From Equation (1), T  [sin(g )w0 0  k t]dt  0 then T sin(kw0 0 t) C   cos(gw0t)dt (19)  0 Adding Equations (15), (19), T T sin( cos(kw0 gw0 t)  2C  cos(gw0  t)dt t)dt 0 0 T T  singw0 (kw0 t    )w0 (20)  t)dt k tdt 0 0 , since the right side of the above equation is zero (see Equation 1). Thus, 2C 0  T
Signals &systems 10EE52  sin( gw0 t) cos(kw0 t)  0 0 for w0  2f f  1 T and k and g are integers. Solution Let T C  gw0 t) cos(kw0t)dt (15) 0 Recall that sin(  )  sin() cos()  sin() cos() (16) Hence, T C   k w0 t sin(kw0 t) cos(gw0 t)dt (17) 0 T T   sin(g  k )w0tdt   sin(kw0 t) cos(gw0 t)dt (18) 0 0 From Equation (1), T  [sin(g  k )w0 t]dt  0 0 then T C  0   sin(kw0 t) cos(gw0t)dt (19) 0 Adding Equations (15), (19), T T 2C   sin( gw0 t) cos(kw0 t)dt  cos(gw0 t)dt 0 0 T T   singw0 t  (kw0 t)dt   k )w0 tdt (20) 0 0 2C  0 , since the right side of the above equation is zero (see Equation 1). Thus, T