What is the value of 8(cos(30)+isin(30)) ?

The value of 8(cos(30)+isin(30)) is 4sqrt(3)+4i

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8(cos(30)+isin(30))

8 (cos (3 0^{\circ}) + i sin (3 0^{\circ}))

43 + 4 i

Solution steps

8 (cos (3 0^{\circ}) + i sin (3 0^{\circ}))

Use the following trivial identity:

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

cos (3 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{3}{2}

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 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{1}{2}

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

Simplify

8 (\frac{3}{2} + i \frac{1}{2}) : 43 + 4 i

8 (\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}

= 8 (\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}

= 8 \cdot \frac{3 + i}{2}

Multiply fractions:

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

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

Divide the numbers:

\frac{8}{2} = 4

= 4 (3 + i)

Rewrite

4 (3 + i)

in standard complex form:

43 + 4 i

4 (3 + i)

Apply the distributive law:

a (b + c) = ab + a c

a = 4, b = 3, c = i

= 43 + 4 i

= 43 + 4 i

= 43 + 4 i

3 \cdot sin (\frac{2 π}{3})

18 sin (7 1^{\circ})

\frac{15}{sin ( 6 8 ^{\circ} )}

arccos (cos (- 3 0^{\circ}))

sin (arcsin (\frac{3 π}{4}))

What is the value of 8(cos(30)+isin(30)) ?
The value of 8(cos(30)+isin(30)) is 4sqrt(3)+4i