Why is ie^(i (pi)/(4))=e^(i (3 pi)/(4))?

Jack Ingram 2022-10-21 Answered
I am doing some matrix multiplication, and at one point it is stated that,
( 1 0 0 e i π 4 ) ( 1 i )   = ( 1 e i 3 π 4 )
but how does i e i π 4 = e i 3 π 4 ?
I have tried taking Euler's formula but I can only get it to reduce down
i 1 2 i
I have also tried converting to polar form, but I just really can't see how these two are equivalent.
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Answers (2)

Martha Dickson
Answered 2022-10-22 Author has 20 answers
This is the whole process of solving:
i e i π 4 = e i π 2 e i π 4 = e i 3 π 4
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Kamila Frye
Answered 2022-10-23 Author has 4 answers
As i 4 = 1, i represents a rotation by 2 π / 4 = π / 2 radians counterclockwise . Similarly, e i π / 4 represents a rotation by another π / 4 radians multiplying represents the composition of the two transformations, for a total of 3 π / 4 radians or e 3 i π / 4
Remember that e i t is a rotation of t radians due to Euler's identity e i t = cos t + i sin t
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