What is the value of sin((8pi)/3) ?

The value of sin((8pi)/3) is (sqrt(3))/2

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sin((8pi)/3)

sin (\frac{8 π}{3})

\frac{3}{2}

Decimal Notation

0.86602 \dots

Solution steps

sin (\frac{8 π}{3})

sin (\frac{8 π}{3}) = sin (\frac{2 π}{3})

sin (\frac{8 π}{3})

Rewrite

\frac{8 π}{3}

2 π + \frac{2 π}{3}

= sin (2 π + \frac{2 π}{3})

Apply the periodicity of

sin

sin (x + 2 π) = sin (x)

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

= sin (\frac{2 π}{3})

= sin (\frac{2 π}{3})

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (\frac{2 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}

= \frac{3}{2}

tan (- \frac{π}{2})

sin (30 0^{\circ})

sin (22 5^{\circ})

2 cos^{2} (θ) - 1 = 0

2 tan^{2} (30 0^{\circ}) + 3 sin^{2} (15 0^{\circ}) - cos^{2} (18 0^{\circ})