What is the value of 4(cos(45)+isin(45)) ?

The value of 4(cos(45)+isin(45)) is 2sqrt(2)+2sqrt(2)i

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4(cos(45)+isin(45))

4 (cos (4 5^{\circ}) + i sin (4 5^{\circ}))

22 + 22 i

Solution steps

4 (cos (4 5^{\circ}) + i sin (4 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}

= 4 (\frac{2}{2} + i \frac{2}{2})

Simplify

4 (\frac{2}{2} + i \frac{2}{2}) : 22 + 22 i

4 (\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}

= 4 (\frac{2 i}{2} + \frac{2}{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}

= 4 \cdot \frac{2 + 2 i}{2}

Multiply fractions:

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

= \frac{( 2 + 2 i ) \cdot 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 (2 + 2 i)

Rewrite

2 (2 + 2 i)

in standard complex form:

22 + 22 i

2 (2 + 2 i)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 2, c = 2 i

= 22 + 22 i

= 22 + 22 i

= 22 + 22 i

arccos (\frac{18}{25})

2 cos (2 - π)

cos (100 0^{\circ})

cos (11 8^{\circ})

- 0 cos (0) + sin (0)

What is the value of 4(cos(45)+isin(45)) ?
The value of 4(cos(45)+isin(45)) is 2sqrt(2)+2sqrt(2)i