What is the value of cos((2pi)/3)+i*sin((2pi)/3) ?

The value of cos((2pi)/3)+i*sin((2pi)/3) is -1/2+i(sqrt(3))/2

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cos((2pi)/3)+i*sin((2pi)/3)

cos (\frac{2 π}{3}) + i \cdot sin (\frac{2 π}{3})

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

Solution steps

cos (\frac{2 π}{3}) + i sin (\frac{2 π}{3})

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 π}{3})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{1}{2}

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (\frac{2 π}{3})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

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

1 + 2 cos (0)

arccos (0.17)

arctan (\frac{- 3}{- 1})

arccos (0.18)

\frac{tan ( 1 6 ^{\circ} ) + tan ( 1 4 ^{\circ} )}{1 - tan ( 1 6 ^{\circ} ) tan ( 1 4 ^{\circ} )}

What is the value of cos((2pi)/3)+i*sin((2pi)/3) ?
The value of cos((2pi)/3)+i*sin((2pi)/3) is -1/2+i(sqrt(3))/2