What is the value of sin^3(1/2 pi) ?

The value of sin^3(1/2 pi) is 1

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sin^3(1/2 pi)

sin^{3} (\frac{1}{2} π)

1

Solution steps

sin^{3} (\frac{1}{2} π)

Rewrite using trig identities:

sin (\frac{1}{2} π) = 1

sin (\frac{1}{2} π)

Simplify:

\frac{1}{2} π = \frac{π}{2}

\frac{1}{2} π

Multiply fractions:

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

= \frac{1 π}{2}

Multiply:

1 π = π

= \frac{π}{2}

= sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

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}

= 1

= 1

= 1^{3}

Simplify

= 1

81 (cos (\frac{3 π}{8}) + i sin (\frac{3 π}{8}))

6 \cdot sin (\frac{π}{4})

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sin (\frac{1}{100})