What is the value of 4(cos(pi/6)+isin(pi/6)) ?

The value of 4(cos(pi/6)+isin(pi/6)) is 2sqrt(3)+2i

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

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

23 + 2 i

Solution steps

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

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{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{π}{6}) = \frac{1}{2}

sin (\frac{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}

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

Simplify

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

4 (\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}

= 4 (\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}

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

Multiply fractions:

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

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

Divide the numbers:

\frac{4}{2} = 2

= 2 (3 + i)

Rewrite

2 (3 + i)

in standard complex form:

23 + 2 i

2 (3 + i)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 3, c = i

= 23 + 2 i

= 23 + 2 i

= 23 + 2 i

arcsin (0.8245)

\frac{- 2 ( - 2 sec ^{2} ( 0 ) + 3 sec ^{4} ( 0 ) )}{cos ( 0 )}

sin (9.4)

sin (sin (9 0^{\circ}))

csc (arctan (\frac{4}{3}))

What is the value of 4(cos(pi/6)+isin(pi/6)) ?
The value of 4(cos(pi/6)+isin(pi/6)) is 2sqrt(3)+2i