The solution to e^{pi/3 i} is 1/2+i(sqrt(3))/2

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simplify e^{pi/3 i}

Solution

simplify

e^{\frac{π}{3} i}

\frac{1}{2} + i \frac{3}{2}

Solution steps

e^{\frac{π}{3} i}

Multiply

\frac{π}{3} i : \frac{πi}{3}

= e^{\frac{πi}{3}}

Apply imaginary number rule:

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

= cos (\frac{π}{3}) + i sin (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

= \frac{1}{2} + i \frac{3}{2}

Multiply

\frac{3}{2} i : \frac{3 i}{2}

= \frac{1}{2} + \frac{3 i}{2}

Apply rule

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

= \frac{1 + 3 i}{2}

Rewrite

\frac{1 + 3 i}{2}

in standard complex form:

\frac{1}{2} + \frac{3}{2} i

= \frac{1}{2} + \frac{3}{2} i

(s + 1) (2 s + 1)

\frac{2}{5} x^{\frac{5}{2}}

3 \frac{3}{4} + 5 \frac{5}{9} + 7 \frac{1}{12}

(- e^{x})^{2}

y^{2} + 6 y - 40