What is the value of (-sqrt(3)+i)(1/8 (cos((5pi)/3)+isin((5pi)/3))) ?

The value of (-sqrt(3)+i)(1/8 (cos((5pi)/3)+isin((5pi)/3))) is i/4

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(-sqrt(3)+i)(1/8 (cos((5pi)/3)+isin((5pi)/3)))

Solution

(- 3 + i) (\frac{1}{8} (cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3})))

Solution

\frac{i}{4}

Solution steps

(- 3 + i) (\frac{1}{8} (cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3})))

= (- 3 + i) \frac{1}{8} (cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3}))

Rewrite using trig identities:

cos (\frac{5 π}{3}) = \frac{1}{2}

cos (\frac{5 π}{3})

Rewrite using trig identities:

cos (π) cos (\frac{2 π}{3}) - sin (π) sin (\frac{2 π}{3})

cos (\frac{5 π}{3})

Write

cos (\frac{5 π}{3})

cos (π + \frac{2 π}{3})

= cos (π + \frac{2 π}{3})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (π) cos (\frac{2 π}{3}) - sin (π) sin (\frac{2 π}{3})

= cos (π) cos (\frac{2 π}{3}) - sin (π) sin (\frac{2 π}{3})

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 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 (π) = 0

sin (π)

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}

= 0

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (\frac{2 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}

= (- 1) (- \frac{1}{2}) - 0 \cdot \frac{3}{2}

Simplify

= \frac{1}{2}

Rewrite using trig identities:

sin (\frac{5 π}{3}) = - \frac{3}{2}

sin (\frac{5 π}{3})

Rewrite using trig identities:

sin (π) cos (\frac{2 π}{3}) + cos (π) sin (\frac{2 π}{3})

sin (\frac{5 π}{3})

Write

sin (\frac{5 π}{3})

sin (π + \frac{2 π}{3})

= sin (π + \frac{2 π}{3})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (π) cos (\frac{2 π}{3}) + cos (π) sin (\frac{2 π}{3})

= sin (π) cos (\frac{2 π}{3}) + cos (π) sin (\frac{2 π}{3})

Use the following trivial identity:

sin (π) = 0

sin (π)

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}

= 0

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 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:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (\frac{2 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}

= 0 \cdot (- \frac{1}{2}) + (- 1) \frac{3}{2}

Simplify

= - \frac{3}{2}

= (- 3 + i) \frac{1}{8} (\frac{1}{2} + i (- \frac{3}{2}))

Simplify

(- 3 + i) \frac{1}{8} (\frac{1}{2} + i (- \frac{3}{2})) : \frac{i}{4}

(- 3 + i) \frac{1}{8} (\frac{1}{2} + i (- \frac{3}{2}))

Remove parentheses:

(- a) = - a

= (- 3 + i) \frac{1}{8} (\frac{1}{2} - i \frac{3}{2})

Multiply fractions:

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

= \frac{1 \cdot ( - 3 + i ) ( \frac{1}{2} - i \frac{3}{2} )}{8}

Multiply:

1 \cdot (- 3 + i) = (- 3 + i)

= \frac{( - 3 + i ) ( - i \frac{3}{2} + \frac{1}{2} )}{8}

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

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

Apply complex arithmetic rule:

(a + bi) (c + d i) = (a c - b d) + (a d + b c) i

a = - 3, b = 1, c = \frac{1}{2}, d = - \frac{3}{2}

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

- 3 \frac{1}{2} - 1 \cdot (- \frac{3}{2}) = 0

- 3 \frac{1}{2} - 1 \cdot (- \frac{3}{2})

Apply rule

- (- a) = a

= - 3 \frac{1}{2} + 1 \cdot \frac{3}{2}

3 \frac{1}{2} = \frac{3}{2}

3 \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 3}{2}

Multiply:

1 \cdot 3 = 3

= \frac{3}{2}

1 \cdot \frac{3}{2} = \frac{3}{2}

1 \cdot \frac{3}{2}

Multiply:

1 \cdot \frac{3}{2} = \frac{3}{2}

= \frac{3}{2}

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

Add similar elements:

- \frac{3}{2} + \frac{3}{2} = 0

= 0

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

- 3 (- \frac{3}{2}) + 1 \cdot \frac{1}{2}

Apply rule

- (- a) = a

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

3 \frac{3}{2} = \frac{3}{2}

3 \frac{3}{2}

Multiply fractions:

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

= \frac{3 3}{2}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{2}

1 \cdot \frac{1}{2} = \frac{1}{2}

1 \cdot \frac{1}{2}

Multiply:

1 \cdot \frac{1}{2} = \frac{1}{2}

= \frac{1}{2}

= \frac{3}{2} + \frac{1}{2}

Apply rule

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

= \frac{3 + 1}{2}

Add the numbers:

3 + 1 = 4

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

= 0 + 2 i

Refine

= 2 i

= \frac{2 i}{8}

Cancel the common factor:

2

= \frac{i}{4}

= \frac{i}{4}

Popular Examples

arccos(2/(sqrt(4)))

arccos (\frac{2}{4})

sin((25pi)/3)

sin (\frac{25 π}{3})

cos(2(pi/3))

cos (2 (\frac{π}{3}))

sin(2sqrt(3))

sin (23)

0.75-cos(0.75)

0.75 - cos (0.75)

Frequently Asked Questions (FAQ)

What is the value of (-sqrt(3)+i)(1/8 (cos((5pi)/3)+isin((5pi)/3))) ?
The value of (-sqrt(3)+i)(1/8 (cos((5pi)/3)+isin((5pi)/3))) is i/4