How do you evaluate # e^( ( 7 pi)/4 i) - e^( ( 3 pi)/2 i)# using trigonometric functions?

Question

1306 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-04-09T18:28:53

#- 0.317 + 0.921i#

According to Euler's theorem,

#e^(ix) = cosx + isinx#

Therefore, using values for #x# from the question,

#x = (7pi)/4#
#e^((7pi)/4i) = cos((7pi)/4) + isin((7pi)/4)#
# = cos315 + isin315#
# = 0.667 + 0.745i#

#x = (3pi)/2#
#e^((3pi)/2i) = cos((3pi)/2) + isin((3pi)/2)#
# = cos270 + isin270#
# = 0.984 - 0.176i#

Put both these values back into the original question,

#e^((7pi)/4i) - e^((3pi)/2i) = 0.667 + 0.745i - 0.984 + 0.176i#
# = - 0.317 + 0.921i#