What is the value of 4sin((2pi)/3)cos((2pi)/3) ?

The value of 4sin((2pi)/3)cos((2pi)/3) is -sqrt(3)

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4sin((2pi)/3)cos((2pi)/3)

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

- 3

Decimal Notation

- 1.73205 \dots

Solution steps

4 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}

= 4 \cdot \frac{3}{2} (- \frac{1}{2})

Simplify

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

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

Remove parentheses:

(- a) = - a

= - 4 \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 4}{2 \cdot 2}

Multiply the numbers:

1 \cdot 4 = 4

= - \frac{4 3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{4 3}{4}

Divide the numbers:

\frac{4}{4} = 1

= - 3

= - 3

cos (arcsin (\frac{1}{2}) + arccot (\frac{3}{3}))

arccos (\frac{59}{95})

cot (- \frac{17 π}{6})

sin (π + \frac{π}{4})

cos (arctan (\frac{4}{3}) + arccos (\frac{12}{13}))

What is the value of 4sin((2pi)/3)cos((2pi)/3) ?
The value of 4sin((2pi)/3)cos((2pi)/3) is -sqrt(3)