How do you evaluate # e^( ( 11 pi)/6 i) - e^( ( 17 pi)/8 i)# using trigonometric functions?

#e^((11pi)/6i)-e^((17pi)/8i)=x+iy#

#e^((11pi)/6i)=cos((11pi)/6)+isin((11pi)/6)#

#e^((17pi)/8i)=cos((17pi)/8)+isin((17pi)/8)#

#e^((11pi)/6i)-e^((17pi)/8i)=(cos((11pi)/6)+isin((11pi)/6))-(cos((17pi)/8)+isin((17pi)/8))#

#e^((11pi)/6i)-e^((17pi)/8i)#
#=(cos((11pi)/6)-cos((17pi)/8))#
#+i(sin((11pi)/6)-sin((17pi)/8))#

#cos((11pi)/6)-cos((17pi)/8)=-2sin(((11pi)/6+(17pi)/8)/2)sin(((11pi)/6-(17pi)/8)/2)#

#(11pi)/6+(17pi)/8=(8xx11pi)/(8xx6)+(6xx17pi)/(6xx8)#

#=(88pi)/48+(102pi)/48=((88+102)pi)/48#

#(11pi)/6+(17pi)/8=(190pi)/48#
#(11pi)/6+(17pi)/8=(95pi)/24#

#(11pi)/6-(17pi)/8=(8xx11pi)/(8xx6)-(6xx17pi)/(6xx8)#

#=(88pi)/48-(102pi)/48=((88-102)pi)/48#

#(11pi)/6-(17pi)/8=(-14pi)/48#
#(11pi)/6-(17pi)/8=(-7pi)/24#

#1/2((11pi)/6+(17pi)/8)=(95pi)/48#

#1/2((11pi)/6-(17pi)/8)=(-7pi)/48#

#-2sin(((11pi)/6+(17pi)/8)/2)sin(((11pi)/6-(17pi)/8)/2)=-2sin((95pi)/48)sin((-7pi)/48)#

#cos((11pi)/6)-cos((17pi)/8)=-2sin((95pi)/48)sin((-7pi)/48)#

#sin((11pi)/6)-sin((17pi)/8)=2cos(((11pi)/6+(17pi)/8)/2)sin(((11pi)/6-(17pi)/8)/2)#

#1/2((11pi)/6+(17pi)/8)=(95pi)/48#

#1/2((11pi)/6-(17pi)/8)=(-7pi)/48#

#sin((11pi)/6)-sin((17pi)/8)=2cos((95pi)/48)sin((-7pi)/48)#

#e^((11pi)/6i)-e^((17pi)/8i)#
#=(cos((11pi)/6)-cos((17pi)/8))#
#+i(sin((11pi)/6)-sin((17pi)/8))#

#cos((11pi)/6)-cos((17pi)/8)=-2sin((95pi)/48)sin((-7pi)/48)#
#sin((11pi)/6)-sin((17pi)/8)=2cos((95pi)/48)sin((-7pi)/48)#

#e^((11pi)/6i)-e^((17pi)/8i)=-2sin((95pi)/48)sin((-7pi)/48)+i(2cos((95pi)/48)sin((-7pi)/48))#

#(95pi)/48=6.218#
#(7pi)/48=0.458#

#cos((95pi)/48)=0.998#
#sin((95pi)/48)=-0.0654#
#cos((-7pi)/48)=0.897#
#sin((-7pi)/48)=-0.442#

#-2(-0.0654)xx(-0.442)+i(2(0.998)xx(-0.442)#

#2xx0.0654xx0.442-2xx0.998xx0.442i#

#e^((11pi)/6i)-e^((17pi)/8i)=0.058-0.882i#

#r=sqrt(0.058^2+(-0.882)^2=0.884#

#theta=tan^-1(-0.882)/(0.058)=-1.505#

#0.058-0.882i=0.884(cos(-1.505)+isin(-1.505))#

#cos(-1.505)+isin(-1.505)=e^((-1.505)i#
#e^((11pi)/6i)-e^((17pi)/8i)=0.884e^(-1.505i)#