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

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

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sin(2((2pi)/3))

sin (2 (\frac{2 π}{3}))

- \frac{3}{2}

Decimal Notation

- 0.86602 \dots

Solution steps

sin (2 (\frac{2 π}{3}))

= sin (2 \cdot \frac{2 π}{3})

Rewrite using trig identities:

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

sin (2 \cdot \frac{2 π}{3})

Use the Double Angle identity:

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

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

= 2 sin (\frac{2 π}{3}) cos (\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}

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 π}{3})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{1}{2}

= 2 \cdot \frac{3}{2} (- \frac{1}{2})

Simplify

2 \cdot \frac{3}{2} (- \frac{1}{2}) : - \frac{3}{2}

2 \cdot \frac{3}{2} (- \frac{1}{2})

Remove parentheses:

(- a) = - a

= - 2 \cdot \frac{3}{2} \cdot \frac{1}{2}

Multiply fractions:

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

= - \frac{3 \cdot 1 \cdot 2}{2 \cdot 2}

Cancel the common factor:

2

= - \frac{3 \cdot 1}{2}

Multiply:

3 \cdot 1 = 3

= - \frac{3}{2}

= - \frac{3}{2}

sin (3 0^{\circ}) \cdot 18

\frac{( sin ^{2} ( 0 ) )}{2}

tan (\frac{37 π}{4})

3.898 \cdot cos (4 5^{\circ})

cos (0.81)

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