What is the value of 3(cos(45)+isin(45))*2(cos(135)+isin(135)) ?

The value of 3(cos(45)+isin(45))*2(cos(135)+isin(135)) is -6

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3(cos(45)+isin(45))*2(cos(135)+isin(135))

Solution

3 (cos (4 5^{\circ}) + i sin (4 5^{\circ})) \cdot 2 (cos (13 5^{\circ}) + i sin (13 5^{\circ}))

Solution

- 6

Solution steps

3 (cos (4 5^{\circ}) + i sin (4 5^{\circ})) \cdot 2 (cos (13 5^{\circ}) + i sin (13 5^{\circ}))

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

Use the following trivial identity:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 5^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{2}{2}

Use the following trivial identity:

sin (13 5^{\circ}) = \frac{2}{2}

sin (13 5^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

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

Simplify

3 (\frac{2}{2} + i \frac{2}{2}) \cdot 2 (- \frac{2}{2} + i \frac{2}{2}) : - 6

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

Multiply

i \frac{2}{2} : \frac{2 i}{2}

i \frac{2}{2}

Multiply fractions:

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

= \frac{2 i}{2}

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

Multiply

i \frac{2}{2} : \frac{2 i}{2}

i \frac{2}{2}

Multiply fractions:

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

= \frac{2 i}{2}

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

Simplify

\frac{2}{2} + \frac{2 i}{2} : \frac{2 + 2 i}{2}

\frac{2}{2} + \frac{2 i}{2}

Apply rule

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

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

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

Combine the fractions

- \frac{2}{2} + \frac{2 i}{2} : \frac{- 2 + 2 i}{2}

Apply rule

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

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

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

Apply complex arithmetic rule:

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

a = 32, b = 32, c = - \frac{2}{2}, d = \frac{2}{2}

= (32 (- \frac{2}{2}) - 32 \frac{2}{2}) + (32 \frac{2}{2} + 32 (- \frac{2}{2})) i

32 (- \frac{2}{2}) - 32 \frac{2}{2} = - 6

32 (- \frac{2}{2}) - 32 \frac{2}{2}

Remove parentheses:

(- a) = - a

= - 32 \frac{2}{2} - 32 \frac{2}{2}

Add similar elements:

- 32 \frac{2}{2} - 32 \frac{2}{2} = - 2 \cdot 32 \frac{2}{2}

= - 2 \cdot 32 \frac{2}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= - 2 \cdot 32

Apply radical rule:

a a = a

22 = 2

= - 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= - 6

32 \frac{2}{2} + 32 (- \frac{2}{2}) = 0

32 \frac{2}{2} + 32 (- \frac{2}{2})

Remove parentheses:

(- a) = - a

= 32 \frac{2}{2} - 32 \frac{2}{2}

Add similar elements:

32 \frac{2}{2} - 32 \frac{2}{2} = 0

= 0

= - 6 + 0 i

Refine

= - 6

= - 6

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Frequently Asked Questions (FAQ)

What is the value of 3(cos(45)+isin(45))*2(cos(135)+isin(135)) ?
The value of 3(cos(45)+isin(45))*2(cos(135)+isin(135)) is -6