What is the value of 4(cos((2pi)/3)+isin((2pi)/3)) ?

The value of 4(cos((2pi)/3)+isin((2pi)/3)) is -2+2sqrt(3)i

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

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

- 2 + 23 i

Solution steps

4 (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}

= 4 (- \frac{1}{2} + i \frac{3}{2})

Simplify

4 (- \frac{1}{2} + i \frac{3}{2}) : - 2 + 23 i

4 (- \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}

= 4 (- \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}

= 4 \cdot \frac{- 1 + 3 i}{2}

Multiply fractions:

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

= \frac{( - 1 + 3 i ) \cdot 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 (- 1 + 3 i)

Rewrite

2 (- 1 + 3 i)

in standard complex form:

- 2 + 23 i

2 (- 1 + 3 i)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = - 1, c = 3 i

= 2 (- 1) + 23 i

Apply minus-plus rules

+ (- a) = - a

= - 2 \cdot 1 + 23 i

Multiply the numbers:

2 \cdot 1 = 2

= - 2 + 23 i

= - 2 + 23 i

= - 2 + 23 i

3 arctan (1)

\frac{2}{tan ( 4 5 ^{\circ} )}

arctan (\frac{- 2}{2})

sec^{2} (\frac{π}{3}) - tan^{2} (\frac{π}{3})

arccos (- 1.82)

What is the value of 4(cos((2pi)/3)+isin((2pi)/3)) ?
The value of 4(cos((2pi)/3)+isin((2pi)/3)) is -2+2sqrt(3)i