How do you multiply e^((11pi )/ 12 ) * e^( pi/4 i ) e11π12eπ4i in trigonometric form?

1 Answer
May 30, 2018

The answer is =-sqrt3/2-1/2i=3212i

Explanation:

Apply Euler's relation

e^(itheta)=costheta+isinthetaeiθ=cosθ+isinθ

e^(11/12ipi)=cos(11/12pi)+isin(11/12pi)e1112iπ=cos(1112π)+isin(1112π)

e^(1/4ipi)=cos(1/4pi)+isin(1/4pi)e14iπ=cos(14π)+isin(14π)

i^2=-1i2=1

Therefore,

e^(11/12ipi)*e^(1/4ipi)=(cos(11/12pi)+isin(11/12pi))*(cos(1/4pi)+isin(1/4pi))e1112iπe14iπ=(cos(1112π)+isin(1112π))(cos(14π)+isin(14π))

=cos(11/12pi)cos(1/4pi)-sin(11/12pi)sin(1/4pi)+i(cos(11/12pi)sin(1/4pi)+sin(11/12pi)cos(1/4pi)))=cos(1112π)cos(14π)sin(1112π)sin(14π)+i(cos(1112π)sin(14π)+sin(1112π)cos(14π)))

=cos(11/12pi+1/4pi)+isin(11/12pi+1/4pi)=cos(1112π+14π)+isin(1112π+14π)

=cos(14/12pi)+isin(14/12pi)=cos(1412π)+isin(1412π)

=cos(7/6pi)+isin(7/6pi)=cos(76π)+isin(76π)

=-sqrt3/2-1/2i=3212i