What is the value of [2(cos(pi/3)+isin(pi/3))]^4 ?

The value of [2(cos(pi/3)+isin(pi/3))]^4 is (1+sqrt(3)i)^4

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[2(cos(pi/3)+isin(pi/3))]^4

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

(1 + 3 i)^{4}

Solution steps

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

= (2 (cos (\frac{π}{3}) + i sin (\frac{π}{3})))^{4}

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

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{1}{2}

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

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{3}{2}

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

Simplify

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

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

Multiply

i \frac{3}{2} : \frac{3 i}{2}

i \frac{3}{2}

Multiply fractions:

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

= \frac{3 i}{2}

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

Combine the fractions

\frac{1}{2} + \frac{3 i}{2} : \frac{1 + 3 i}{2}

Apply rule

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

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

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

Remove parentheses:

(a) = a

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

Multiply

2 \cdot \frac{1 + 3 i}{2} : 1 + 3 i

2 \cdot \frac{1 + 3 i}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1 + 3 i

= (1 + 3 i)^{4}

= (1 + 3 i)^{4}

\frac{5}{sin ( 4 0 ^{\circ} )}

sin (arcsin (\frac{5}{13}) + arccos (- \frac{7}{25}))

cos (π) π

2 tan (- 2)

300 sin (2 0^{\circ})

What is the value of [2(cos(pi/3)+isin(pi/3))]^4 ?
The value of [2(cos(pi/3)+isin(pi/3))]^4 is (1+sqrt(3)i)^4