What is the value of isin(pi/3) ?

The value of isin(pi/3) is (sqrt(3)i)/2

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isin(pi/3)

i sin (\frac{π}{3})

\frac{3 i}{2}

Solution steps

i sin (\frac{π}{3})

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}

= i \frac{3}{2}

Simplify

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}

= \frac{3 i}{2}

sin^{2} (\frac{4 π}{3})

\frac{40 \cdot sin ( 3 7 ^{\circ} )}{30}

cot (\frac{2}{2})

(sin (\frac{π}{3}))^{2}

\frac{23}{cos ( 6 3 ^{\circ} )}