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

The value of 5(cos(pi/6)+isin(pi/6)) is (5sqrt(3))/2+i 5/2

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

\frac{5 3}{2} + i \frac{5}{2}

Solution steps

5 (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}

= 5 (\frac{3}{2} + i \frac{1}{2})

Simplify

5 (\frac{3}{2} + i \frac{1}{2}) : \frac{5 3}{2} + i \frac{5}{2}

5 (\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}

= 5 (\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}

= 5 \cdot \frac{3 + i}{2}

Multiply fractions:

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

= \frac{( 3 + i ) \cdot 5}{2}

Rewrite

\frac{( 3 + i ) \cdot 5}{2}

in standard complex form:

\frac{5 3}{2} + \frac{5}{2} i

\frac{( 3 + i ) \cdot 5}{2}

Expand

(3 + i) \cdot 5 : 53 + 5 i

(3 + i) \cdot 5

= 5 (3 + i)

Apply the distributive law:

a (b + c) = ab + a c

a = 5, b = 3, c = i

= 53 + 5 i

= \frac{5 3 + 5 i}{2}

Apply the fraction rule:

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

\frac{5 3 + 5 i}{2} = \frac{5 3}{2} + \frac{5 i}{2}

= \frac{5 3}{2} + \frac{5 i}{2}

= \frac{5 3}{2} + \frac{5}{2} i

= \frac{5 3}{2} + i \frac{5}{2}

What is the value of 5(cos(pi/6)+isin(pi/6)) ?
The value of 5(cos(pi/6)+isin(pi/6)) is (5sqrt(3))/2+i 5/2