What is the value of 2(cos(120)+isin(120)) ?

The value of 2(cos(120)+isin(120)) is -1+sqrt(3)i

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2(cos(120)+isin(120))

2 (cos (12 0^{\circ}) + i sin (12 0^{\circ}))

- 1 + 3 i

Solution steps

2 (cos (12 0^{\circ}) + i sin (12 0^{\circ}))

Use the following trivial identity:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (12 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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 (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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

- csc (\frac{3 π}{4})

(cos (\frac{π}{4}) + sin (\frac{π}{6}))^{2}

cos (94 5^{\circ})

cot (1 3^{\circ})

cos (arctan (- 5))

What is the value of 2(cos(120)+isin(120)) ?
The value of 2(cos(120)+isin(120)) is -1+sqrt(3)i