What is the value of 6(cos((5pi)/6)+isin((5pi)/6)) ?

The value of 6(cos((5pi)/6)+isin((5pi)/6)) is -3sqrt(3)+3i

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6(cos((5pi)/6)+isin((5pi)/6))

6 (cos (\frac{5 π}{6}) + i sin (\frac{5 π}{6}))

- 33 + 3 i

Solution steps

6 (cos (\frac{5 π}{6}) + i sin (\frac{5 π}{6}))

Use the following trivial identity:

cos (\frac{5 π}{6}) = - \frac{3}{2}

cos (\frac{5 π}{6})

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{3}{2}

Use the following trivial identity:

sin (\frac{5 π}{6}) = \frac{1}{2}

sin (\frac{5 π}{6})

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{1}{2}

= 6 (- \frac{3}{2} + i \frac{1}{2})

Simplify

6 (- \frac{3}{2} + i \frac{1}{2}) : - 33 + 3 i

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

i \frac{1}{2} = \frac{i}{2}

i \frac{1}{2}

Multiply fractions:

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

= \frac{1 i}{2}

Multiply:

1 i = i

= \frac{i}{2}

= 6 (- \frac{3}{2} + \frac{i}{2})

Simplify

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

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

Apply rule

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

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

= 6 \cdot \frac{- 3 + i}{2}

Multiply fractions:

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

= \frac{( - 3 + i ) \cdot 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 (- 3 + i)

Rewrite

3 (- 3 + i)

in standard complex form:

- 33 + 3 i

3 (- 3 + i)

Apply the distributive law:

a (b + c) = ab + a c

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

= 3 (- 3) + 3 i

Apply minus-plus rules

+ (- a) = - a

= - 33 + 3 i

= - 33 + 3 i

= - 33 + 3 i

sin (- 1 (0))

cos (- 51 0^{\circ})

sin (arcsin (\frac{12}{13}) + arctan (\frac{3}{4}))

\frac{3}{2} sin^{2} (0)

ln (sin (1))

What is the value of 6(cos((5pi)/6)+isin((5pi)/6)) ?
The value of 6(cos((5pi)/6)+isin((5pi)/6)) is -3sqrt(3)+3i