What is the value of 7(cos(120)+isin(120)) ?

The value of 7(cos(120)+isin(120)) is -7/2+i(7sqrt(3))/2

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7(cos(120)+isin(120))

7 (cos (12 0^{\circ}) + i sin (12 0^{\circ}))

- \frac{7}{2} + i \frac{7 3}{2}

Solution steps

7 (cos (12 0^{\circ}) + i sin (12 0^{\circ}))

Use the following trivial identity:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (12 0^{\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{1}{2}

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\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{3}{2}

= 7 (- \frac{1}{2} + i \frac{3}{2})

Simplify

7 (- \frac{1}{2} + i \frac{3}{2}) : - \frac{7}{2} + i \frac{7 3}{2}

7 (- \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}

= 7 (- \frac{1}{2} + \frac{3 i}{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}

= 7 \cdot \frac{- 1 + 3 i}{2}

Multiply fractions:

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

= \frac{( - 1 + 3 i ) \cdot 7}{2}

Rewrite

\frac{( - 1 + 3 i ) \cdot 7}{2}

in standard complex form:

- \frac{7}{2} + \frac{7 3}{2} i

\frac{( - 1 + 3 i ) \cdot 7}{2}

Expand

(- 1 + 3 i) \cdot 7 : - 7 + 73 i

(- 1 + 3 i) \cdot 7

= 7 (- 1 + 3 i)

Apply the distributive law:

a (b + c) = ab + a c

a = 7, b = - 1, c = 3 i

= 7 (- 1) + 73 i

Apply minus-plus rules

+ (- a) = - a

= - 7 \cdot 1 + 73 i

Multiply the numbers:

7 \cdot 1 = 7

= - 7 + 73 i

= \frac{- 7 + 7 3 i}{2}

Apply the fraction rule:

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

\frac{- 7 + 7 3 i}{2} = - \frac{7}{2} + \frac{7 3 i}{2}

= - \frac{7}{2} + \frac{7 3 i}{2}

= - \frac{7}{2} + \frac{7 3}{2} i

= - \frac{7}{2} + i \frac{7 3}{2}

90 tan (6 3^{\circ})

arcsin (\frac{1}{7})

cos (2 5^{\circ}) cos (5^{\circ}) - sin (2 5^{\circ}) sin (5^{\circ})

\frac{cos ( 5 π )}{12}

arctan (\frac{1.5}{6})

What is the value of 7(cos(120)+isin(120)) ?
The value of 7(cos(120)+isin(120)) is -7/2+i(7sqrt(3))/2