How do you evaluate # e^( ( pi)/4 i) - e^( ( 5 pi)/3 i)# using trigonometric functions?

1 Answer
1s2s2p
Oct 16, 2017

#e^( ( pi)/4 i) - e^( ( 5 pi)/3 i)=(sqrt(2)-1)/2+i(3sqrt(2)+2sqrt(3))/6#

Explanation:

Using Euler's identity:
#e^(ix)=cosx+isinx#

So, #e^( ( pi)/4 i) - e^( ( 5 pi)/3 i)=(cos(\pi/4)+isin(\pi/4))-(cos((5\pi)/3)+isin((5\pi)/3)#
#=(sqrt(2)/2+isqrt(2)/2)-(1/2-isqrt(3)/3)#
#=sqrt(2)/2+isqrt(2)/2-1/2+isqrt(3)/3#
#=sqrt(2)/2+isqrt(2)/2-1/2+isqrt(3)/3#
#=(sqrt(2)-1)/2+i(3sqrt(2)+2sqrt(3))/6#

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