What is the value of 2(sin((4pi)/6))/(1+(4pi)/6) ?

The value of 2(sin((4pi)/6))/(1+(4pi)/6) is (3sqrt(3))/(3+2pi)

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2(sin((4pi)/6))/(1+(4pi)/6)

2 \frac{sin ( \frac{4 π}{6} )}{1 + \frac{4 π}{6}}

\frac{3 3}{3 + 2 π}

Decimal Notation

0.55973 \dots

Solution steps

2 \cdot \frac{sin ( \frac{4 π}{6} )}{1 + \frac{4 π}{6}}

Rewrite using trig identities:

sin (\frac{4 π}{6}) = \frac{3}{2}

sin (\frac{4 π}{6})

Simplify:

\frac{4 π}{6} = \frac{2 π}{3}

\frac{4 π}{6}

Cancel the common factor:

2

= \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}

= 2 \cdot \frac{\frac{3}{2}}{1 + \frac{4 π}{6}}

Simplify

2 \cdot \frac{\frac{3}{2}}{1 + \frac{4 π}{6}} : \frac{3 3}{3 + 2 π}

2 \cdot \frac{\frac{3}{2}}{1 + \frac{4 π}{6}}

\frac{\frac{3}{2}}{1 + \frac{4 π}{6}} = \frac{3}{2 \cdot \frac{3 + 2 π}{3}}

\frac{\frac{3}{2}}{1 + \frac{4 π}{6}}

Apply the fraction rule:

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

= \frac{3}{2 ( 1 + \frac{4 π}{6} )}

Cancel

\frac{4 π}{6} : \frac{2 π}{3}

\frac{4 π}{6}

Cancel the common factor:

2

= \frac{2 π}{3}

= \frac{3}{2 ( \frac{2 π}{3} + 1 )}

Join

1 + \frac{2 π}{3} : \frac{3 + 2 π}{3}

1 + \frac{2 π}{3}

Convert element to fraction:

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

= \frac{1 \cdot 3}{3} + \frac{2 π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 3 + 2 π}{3}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3 + 2 π}{3}

= \frac{3}{2 \cdot \frac{2 π + 3}{3}}

= 2 \cdot \frac{3}{2 \cdot \frac{2 π + 3}{3}}

Multiply fractions:

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

= \frac{3 \cdot 2}{2 \cdot \frac{3 + 2 π}{3}}

Cancel the common factor:

2

= \frac{3}{\frac{3 + 2 π}{3}}

Apply the fraction rule:

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

= \frac{3 \cdot 3}{3 + 2 π}

= \frac{3 3}{3 + 2 π}

- sin (2^{4}) \cdot 4 \cdot 2^{3}

sec (3 0^{\circ}) + sin (6 0^{\circ})

arccos (- \frac{5}{4})

\frac{0 ^{2} - 1}{cos ( 0 )}

sin (\frac{180}{2})

What is the value of 2(sin((4pi)/6))/(1+(4pi)/6) ?
The value of 2(sin((4pi)/6))/(1+(4pi)/6) is (3sqrt(3))/(3+2pi)