What is the value of (cos(pi/4)+isin(pi/4))^3 ?

The value of (cos(pi/4)+isin(pi/4))^3 is -(sqrt(2))/2+i(sqrt(2))/2

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(cos(pi/4)+isin(pi/4))^3

Solution

(cos (\frac{π}{4}) + i sin (\frac{π}{4}))^{3}

Solution

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

Solution steps

(cos (\frac{π}{4}) + i sin (\frac{π}{4}))^{3}

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}

= (\frac{2}{2} + i \frac{2}{2})^{3}

Simplify

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

(\frac{2}{2} + i \frac{2}{2})^{3}

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}

= (\frac{2}{2} + \frac{2 i}{2})^{3}

Combine the fractions

\frac{2}{2} + \frac{2 i}{2} : \frac{2 + 2 i}{2}

Apply rule

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

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

= (\frac{2 + 2 i}{2})^{3}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 2 + 2 i ) ^{3}}{2 ^{3}}

(2 + 2 i)^{3} = 42 i - 42

(2 + 2 i)^{3}

Apply Perfect Cube Formula:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

a = 2, b = 2 i

= (2)^{3} + 3 (2)^{2} 2 i + 32 (2 i)^{2} + (2 i)^{3}

Simplify

(2)^{3} + 3 (2)^{2} 2 i + 32 (2 i)^{2} + (2 i)^{3} : 42 i - 42

(2)^{3} + 3 (2)^{2} 2 i + 32 (2 i)^{2} + (2 i)^{3}

(2)^{3} = 22

(2)^{3}

Apply radical rule:

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

= (2^{\frac{1}{2}})^{3}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 3}

\frac{1}{2} \cdot 3 = \frac{3}{2}

\frac{1}{2} \cdot 3

Multiply fractions:

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

= \frac{1 \cdot 3}{2}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{2}

= 2^{\frac{3}{2}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 22

= 22

3 (2)^{2} 2 i = 62 i

3 (2)^{2} 2 i

(2)^{2} = 2

(2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 3 \cdot 22 i

Multiply the numbers:

3 \cdot 2 = 6

= 62 i

32 (2 i)^{2} = - 62

32 (2 i)^{2}

(2 i)^{2} = 2 i^{2}

(2 i)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{2} (2)^{2}

(2)^{2} : 2

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 2 i^{2}

= 3 \cdot 22 i^{2}

Multiply the numbers:

3 \cdot 2 = 6

= 62 i^{2}

i^{2} = - 1

i^{2}

Apply imaginary number rule:

i^{2} = - 1

= - 1

= 6 (- 1) 2

Remove parentheses:

(- a) = - a

= - 62 \cdot 1

Multiply the numbers:

6 \cdot 1 = 6

= - 62

= 22 + 62 i - 62 + (2 i)^{3}

Group like terms

= 62 i + (2 i)^{3} + 22 - 62

Add similar elements:

22 - 62 = - 42

= 62 i + (2 i)^{3} - 42

(2 i)^{3} = - 22 i

(2 i)^{3}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{3} (2)^{3}

(2)^{3} : 2^{\frac{3}{2}}

Apply radical rule:

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

= (2^{\frac{1}{2}})^{3}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 3}

\frac{1}{2} \cdot 3 = \frac{3}{2}

\frac{1}{2} \cdot 3

Multiply fractions:

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

= \frac{1 \cdot 3}{2}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{2}

= 2^{\frac{3}{2}}

= 2^{\frac{3}{2}} i^{3}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 22

= 22 i^{3}

i^{3} = - i

i^{3}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

i^{3} = i^{2} i

= i^{2} i

Apply imaginary number rule:

i^{2} = - 1

= - 1 i

Multiply:

1 i = i

= - i

= 22 (- i)

Remove parentheses:

(- a) = - a

= - 22 i

= 62 i - 22 i - 42

Add similar elements:

62 i - 22 i = 42 i

= 42 i - 42

= 42 i - 42

= \frac{4 2 i - 4 2}{2 ^{3}}

Factor

42 i - 42 : 42 (i - 1)

42 i - 42

Rewrite as

= 42 i - 1 \cdot 42

Factor out common term

42

= 42 (i - 1)

= \frac{4 2 ( i - 1 )}{2 ^{3}}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

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

\frac{2 ^{2} 2 ( i - 1 )}{2 ^{3}}

Apply exponent rule:

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

\frac{2 ^{2}}{2 ^{3}} = \frac{1}{2 ^{3 - 2}}

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

Subtract the numbers:

3 - 2 = 1

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

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

Rewrite

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

in standard complex form:

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

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

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

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

Apply the fraction rule:

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

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

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

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

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

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

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

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

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

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

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

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Frequently Asked Questions (FAQ)

What is the value of (cos(pi/4)+isin(pi/4))^3 ?
The value of (cos(pi/4)+isin(pi/4))^3 is -(sqrt(2))/2+i(sqrt(2))/2