What is the value of 5(cos(135)+isin(135)) ?

The value of 5(cos(135)+isin(135)) is -(5sqrt(2))/2+i(5sqrt(2))/2

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5(cos(135)+isin(135))

Solution

5 (cos (13 5^{\circ}) + i sin (13 5^{\circ}))

Solution

- \frac{5 2}{2} + i \frac{5 2}{2}

Solution steps

5 (cos (13 5^{\circ}) + i sin (13 5^{\circ}))

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}

= 5 (- \frac{2}{2} + i \frac{2}{2})

Simplify

5 (- \frac{2}{2} + i \frac{2}{2}) : - \frac{5 2}{2} + i \frac{5 2}{2}

5 (- \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}

= 5 (- \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}

= 5 \cdot \frac{- 2 + 2 i}{2}

Multiply fractions:

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

= \frac{( - 2 + 2 i ) \cdot 5}{2}

Factor

- 2 + 2 i : 2 (- 1 + i)

- 2 + 2 i

Factor out common term

2

= 2 (- 1 + i)

= \frac{2 ( - 1 + i ) \cdot 5}{2}

Cancel

\frac{2 ( - 1 + i ) \cdot 5}{2} : \frac{( - 1 + i ) \cdot 5}{2}

\frac{2 ( - 1 + i ) \cdot 5}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{5 \cdot 2 ^{\frac{1}{2}} ( - 1 + i )}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{5 ( - 1 + i )}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{5 ( - 1 + i )}{2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{5 ( - 1 + i )}{2}

= \frac{( - 1 + i ) \cdot 5}{2}

Rationalize

\frac{5 ( - 1 + i )}{2} : \frac{5 2 ( - 1 + i )}{2}

\frac{5 ( - 1 + i )}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{( - 1 + i ) \cdot 5 2}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{5 2 ( - 1 + i )}{2}

= \frac{5 2 ( - 1 + i )}{2}

Rewrite

\frac{5 2 ( - 1 + i )}{2}

in standard complex form:

- \frac{5 2}{2} + \frac{5 2}{2} i

\frac{5 2 ( - 1 + i )}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{5 \cdot 2 ^{\frac{1}{2}} ( - 1 + i )}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{5 ( - 1 + i )}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{5 ( - 1 + i )}{2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{5 ( - 1 + i )}{2}

Expand

5 (- 1 + i) : - 5 + 5 i

5 (- 1 + i)

Apply the distributive law:

a (b + c) = ab + a c

a = 5, b = - 1, c = i

= 5 (- 1) + 5 i

Apply minus-plus rules

+ (- a) = - a

= - 5 \cdot 1 + 5 i

Multiply the numbers:

5 \cdot 1 = 5

= - 5 + 5 i

= \frac{- 5 + 5 i}{2}

Apply the fraction rule:

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

\frac{- 5 + 5 i}{2} = - \frac{5}{2} + \frac{5 i}{2}

= - \frac{5}{2} + \frac{5 i}{2}

\frac{5}{2} = \frac{5 2}{2}

\frac{5}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{5 2}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{5 2}{2}

= - \frac{5}{2} + \frac{5 2}{2} i

- \frac{5}{2} = - \frac{5 2}{2}

- \frac{5}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{5 2}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{5 2}{2}

= - \frac{5 2}{2} + \frac{5 2}{2} i

= - \frac{5 2}{2} + \frac{5 2}{2} i

= - \frac{5 2}{2} + i \frac{5 2}{2}

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Frequently Asked Questions (FAQ)

What is the value of 5(cos(135)+isin(135)) ?
The value of 5(cos(135)+isin(135)) is -(5sqrt(2))/2+i(5sqrt(2))/2