The solution to e^{i pi/4} is (sqrt(2))/2+i(sqrt(2))/2

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simplify e^{i pi/4}

Solution

simplify

e^{i \frac{π}{4}}

\frac{2}{2} + i \frac{2}{2}

Solution steps

e^{i \frac{π}{4}}

Multiply

i \frac{π}{4} : \frac{πi}{4}

= e^{\frac{πi}{4}}

Apply imaginary number rule:

e^{ia} = cos (a) + i sin (a)

= cos (\frac{π}{4}) + i sin (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

= \frac{2}{2} + i \frac{2}{2}

Multiply

\frac{2}{2} i : \frac{2 i}{2}

= \frac{2}{2} + \frac{2 i}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 + 2 i}{2}

Rewrite

\frac{2 + 2 i}{2}

in standard complex form:

\frac{2}{2} + \frac{2}{2} i

= \frac{2}{2} + \frac{2}{2} i

32 x^{2} - 18 y^{2}

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