What is the value of 2(cos((2pi)/3)+isin((2pi)/3)) ?

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

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2(cos((2pi)/3)+isin((2pi)/3))

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

- 1 + 3 i

Solution steps

2 (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}

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

Simplify

2 (- \frac{1}{2} + i \frac{3}{2}) : - 1 + 3 i

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

Multiply

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

i \frac{3}{2}

Multiply fractions:

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

= \frac{3 i}{2}

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

Simplify

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

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

Multiply fractions:

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

= \frac{( - 1 + 3 i ) \cdot 2}{2}

Cancel the common factor:

2

= - - 1 + 3 i

= - 1 + 3 i

\frac{8.3}{tan ( 1 6 ^{\circ} )}

cos (3 0^{\circ} + 4 5^{\circ})

9 sin (6 0^{\circ})

3 sin (- \frac{π}{2})

1 cos (4 5^{\circ})

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