What is the value of 5(cos(pi/4)+isin(pi/4)) ?

The value of 5(cos(pi/4)+isin(pi/4)) is (5sqrt(2))/2+i(5sqrt(2))/2

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5(cos(pi/4)+isin(pi/4))

Solution

5 (cos (\frac{π}{4}) + i sin (\frac{π}{4}))

Solution

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

Solution steps

5 (cos (\frac{π}{4}) + i sin (\frac{π}{4}))

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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 (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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 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}

= 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 \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(pi/4)+isin(pi/4)) ?
The value of 5(cos(pi/4)+isin(pi/4)) is (5sqrt(2))/2+i(5sqrt(2))/2