What is the value of sin(4(4pi)/3) ?

The value of sin(4(4pi)/3) is sin((4pi)/3)

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sin(4(4pi)/3)

sin (4 \frac{4 π}{3})

sin (\frac{4 π}{3})

Decimal Notation

- 0.86602 \dots

Solution steps

sin (4 \cdot \frac{4 π}{3})

Rewrite as

= sin (2 \cdot \frac{8 π}{3})

Rewrite using trig identities:

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

sin (2 \cdot \frac{8 π}{3})

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (\frac{8 π}{3}) cos (\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})

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

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

cos (\frac{8 π}{3})

Rewrite

\frac{8 π}{3}

2 π + \frac{2 π}{3}

= cos (2 π + \frac{2 π}{3})

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

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

= cos (\frac{2 π}{3})

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

= sin (4 \cdot \frac{4 π}{3})

Simplify:

4 \cdot \frac{4 π}{3} = \frac{16 π}{3}

4 \cdot \frac{4 π}{3}

Multiply fractions:

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

= \frac{4 π 4}{3}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{16 π}{3}

= sin (\frac{16 π}{3})

sin (\frac{16 π}{3}) = sin (\frac{4 π}{3})

sin (\frac{16 π}{3})

Rewrite

\frac{16 π}{3}

2 π \cdot 2 + \frac{4 π}{3}

= sin (2 π 2 + \frac{4 π}{3})

Apply the periodicity of

sin

sin (x + 2 π \cdot k) = sin (x)

sin (2 π \cdot 2 + \frac{4 π}{3}) = sin (\frac{4 π}{3})

= sin (\frac{4 π}{3})

= sin (\frac{4 π}{3})

sin (14 8^{\circ})

cos (\frac{2 π}{3} + \frac{π}{4})

cos^{2} (6 0^{\circ}) - cos^{2} (3 0^{\circ})

1 + 2 cos (36 0^{\circ})

cos^{2} (60.2 6^{\circ}) \cdot tan (60.2 6^{\circ})