What is the value of 27(cos(pi/3)+isin(pi/3)) ?

The value of 27(cos(pi/3)+isin(pi/3)) is 27/2+i(27sqrt(3))/2

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

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

\frac{27}{2} + i \frac{27 3}{2}

Solution steps

27 (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}

= 27 (\frac{1}{2} + i \frac{3}{2})

Simplify

27 (\frac{1}{2} + i \frac{3}{2}) : \frac{27}{2} + i \frac{27 3}{2}

27 (\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}

= 27 (\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}

= 27 \cdot \frac{1 + 3 i}{2}

Multiply fractions:

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

= \frac{( 1 + 3 i ) \cdot 27}{2}

Rewrite

\frac{( 1 + 3 i ) \cdot 27}{2}

in standard complex form:

\frac{27}{2} + \frac{27 3}{2} i

\frac{( 1 + 3 i ) \cdot 27}{2}

Expand

(1 + 3 i) \cdot 27 : 27 + 273 i

(1 + 3 i) \cdot 27

= 27 (1 + 3 i)

Apply the distributive law:

a (b + c) = ab + a c

a = 27, b = 1, c = 3 i

= 27 \cdot 1 + 273 i

Multiply the numbers:

27 \cdot 1 = 27

= 27 + 273 i

= \frac{27 + 27 3 i}{2}

Apply the fraction rule:

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

\frac{27 + 27 3 i}{2} = \frac{27}{2} + \frac{27 3 i}{2}

= \frac{27}{2} + \frac{27 3 i}{2}

= \frac{27}{2} + \frac{27 3}{2} i

= \frac{27}{2} + i \frac{27 3}{2}

1 + 0.1 \cdot (\frac{1.2}{2}) \cdot tan^{2} (6 0^{\circ})

e^{\frac{π}{2}} cos (\frac{π}{2})

cos (0.86)

cos (0.83)

cos (\frac{39 π}{6})

What is the value of 27(cos(pi/3)+isin(pi/3)) ?
The value of 27(cos(pi/3)+isin(pi/3)) is 27/2+i(27sqrt(3))/2