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

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

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sin(2(3pi)/2)

sin (2 \frac{3 π}{2})

0

Solution steps

sin (2 \cdot \frac{3 π}{2})

Simplify:

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

= sin (3 π)

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)

= sin (π)

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

= 0

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

3 arctan (1)

\frac{2}{tan ( 4 5 ^{\circ} )}

arctan (\frac{- 2}{2})

sec^{2} (\frac{π}{3}) - tan^{2} (\frac{π}{3})