# 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,

but how does $i{e}^{i\frac{\pi }{4}}={e}^{i\frac{3\pi }{4}}$?
I have tried taking Euler's formula but I can only get it to reduce down
$\frac{-i-1}{\sqrt{2}i}$
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:
$i{e}^{i\frac{\pi }{4}}={e}^{i\frac{\pi }{2}}{e}^{i\frac{\pi }{4}}={e}^{i\frac{3\pi }{4}}$
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Kamila Frye
Answered 2022-10-23 Author has 4 answers
As ${i}^{4}=1$, i represents a rotation by $2\pi /4=\pi /2$ radians counterclockwise . Similarly, ${e}^{i\pi /4}$ represents a rotation by another $\pi /4$ radians multiplying represents the composition of the two transformations, for a total of $3\pi /4$ radians or ${e}^{3i\pi /4}$
Remember that ${e}^{it}$ is a rotation of t radians due to Euler's identity ${e}^{it}=\mathrm{cos}t+i\mathrm{sin}t$
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