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

The value of 1/2 sin(2 3/2 pi) is 0

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1/2 sin(2 3/2 pi)

\frac{1}{2} sin (2 \frac{3}{2} π)

0

Solution steps

\frac{1}{2} sin (2 \cdot \frac{3}{2} π)

Simplify:

2 \cdot \frac{3}{2} π = 3 π

2 \cdot \frac{3}{2} π

Multiply fractions:

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

= \frac{3 \cdot 2 π}{2}

Cancel the common factor:

2

= 3 π

\frac{1}{2} sin (3 π) = \frac{sin ( 3 π )}{2}

\frac{1}{2} sin (3 π)

Multiply fractions:

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

= \frac{1 \cdot sin ( 3 π )}{2}

Multiply:

1 \cdot sin (3 π) = sin (3 π)

= \frac{sin ( 3 π )}{2}

= \frac{sin ( 3 π )}{2}

sin (3 π) = sin (π)

sin (3 π)

Rewrite

3 π

2 π + π

= sin (2 π + π)

Apply the periodicity of

sin

sin (x + 2 π) = sin (x)

sin (2 π + π) = sin (π)

= sin (π)

= \frac{sin ( π )}{2}

Use the following trivial identity:

sin (π) = 0

sin (π)

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}

= 0

= \frac{0}{2}

Simplify

= 0

cos (3 0^{\circ}) + cos (6 0^{\circ})

- 2 sin (- \frac{π}{2}) - 1

sin (233.1 3^{\circ})

sin (123 0^{\circ})

2 sin (\frac{1}{2}) π