How do you evaluate # e^( ( 7 pi)/4 i) - e^( ( 8 pi)/3 i)# using trigonometric functions?

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Answer 1 · 2016-05-03T09:23:22

#0.881-1.692i#

According to Euler's formula,

#e^(ix)=cosx+isinx#

If we substitute in values for #x# from the question, we get

#e^((7pi)/4i)=cos((7pi)/4)+isin((7pi)/4)#
#=cos315+isin315#
#=0.707-0.707i#

#e^((8pi)/3i)=cos((8pi)/3)+isin((8pi)/3)#
#=cos100+isin100#
#=-0.174+0.985i#

Now you have the two parts of the question, you put them together and solve arithmetically:

#e^((7pi)/4i)-e^((8pi)/3i)#
#=(0.707-0.707i)-(-0.174+0.985i)#
#=0.707+0.174-0.707i-0.985i#

#=0.881-1.692i#