Question

If a = cosx+isinx, b= cosy+isiny find (i) ab

Answer 1

$by eulers formula, we have</p><p></p><p>a=e^ix and b=e^iy</p><p></p><p>now, a*b=e^i(x+y)=cos(x+y)+isin(x+y)</p><p></p><p>and 1/ab=e^-i(x+y)=cos(x+y) – isin(x+y) (because cos( – z)=cosz and sin( – z)= – sinz)</p><p></p><p>adding, we get </p><p></p><p>cos(x+y)= (ab+1/ab)/2</p><p></p><p>now, a/b=e^i(x-y)=cos(x-y)+isin(x-y)</p><p></p><p>and b/a=e^i(y-x)=cos(x-y) – isin(x-y) (because cos( – z)=cosz and sin( – z)= – sinz)</p><p></p><p>adding, we have</p><p></p><p>cos(x-y)= (a/b+b/a)/2</p><p></p><p>$