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

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simplify e^{i pi/3}

Solution

simplify

e^{i \frac{π}{3}}

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

Solution steps

e^{i \frac{π}{3}}

Multiply

i \frac{π}{3} : \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

\frac{3 a ^{2} - 24 a}{3 a ^{2} + 12 a}

56 - c \geq 54

0 = 0.01658 x^{2} - 21.489 x + 6839.39

3 (2 x - 1) \leq \frac{1}{4} (2 x + 8)

\frac{x ^{3} + 3 x ^{2} - 9 x - 27}{x + 4} \leq 0