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

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

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

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

2 + 23 i

Solution steps

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

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{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{π}{3}) = \frac{3}{2}

sin (\frac{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{3 i}{2} + \frac{1}{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 \cdot 1 + 23 i

Multiply the numbers:

2 \cdot 1 = 2

= 2 + 23 i

= 2 + 23 i

= 2 + 23 i

sec (\frac{5}{4} π)

cot (3 3^{\circ})

cos (2 π) + 1

sin (\frac{π}{4}) + tan (\frac{π}{6})

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

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