What is the value of (2(1+sin(pi/3)))/(2(pi/3-cos(pi/3))) ?

The value of (2(1+sin(pi/3)))/(2(pi/3-cos(pi/3))) is (3(2+sqrt(3)))/(2pi-3)

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(2(1+sin(pi/3)))/(2(pi/3-cos(pi/3)))

Solution

\frac{2 ( 1 + sin ( \frac{π}{3} ) )}{2 ( \frac{π}{3} - cos ( \frac{π}{3} ) )}

Solution

\frac{3 ( 2 + 3 )}{2 π - 3}

Decimal Notation

3.41014 \dots

Solution steps

\frac{2 ( 1 + sin ( \frac{π}{3} ) )}{2 ( \frac{π}{3} - cos ( \frac{π}{3} ) )}

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{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{π}{3}) = \frac{1}{2}

cos (\frac{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}

= \frac{2 ( 1 + \frac{3}{2} )}{2 ( \frac{π}{3} - \frac{1}{2} )}

Simplify

\frac{2 ( 1 + \frac{3}{2} )}{2 ( \frac{π}{3} - \frac{1}{2} )} : \frac{3 ( 2 + 3 )}{2 π - 3}

\frac{2 ( 1 + \frac{3}{2} )}{2 ( \frac{π}{3} - \frac{1}{2} )}

Divide the numbers:

\frac{2}{2} = 1

= \frac{1 + \frac{3}{2}}{( \frac{π}{3} - \frac{1}{2} )}

Remove parentheses:

(a) = a

= \frac{1 + \frac{3}{2}}{\frac{π}{3} - \frac{1}{2}}

Join

\frac{π}{3} - \frac{1}{2} : \frac{2 π - 3}{6}

\frac{π}{3} - \frac{1}{2}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

For

\frac{1}{2} :

multiply the denominator and numerator by

3

\frac{1}{2} = \frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}

= \frac{π 2}{6} - \frac{3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - 3}{6}

= \frac{1 + \frac{3}{2}}{\frac{2 π - 3}{6}}

Join

1 + \frac{3}{2} : \frac{2 + 3}{2}

1 + \frac{3}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 + 3}{2}

= \frac{\frac{2 + 3}{2}}{\frac{2 π - 3}{6}}

Divide fractions:

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

= \frac{( 2 + 3 ) \cdot 6}{2 ( π 2 - 3 )}

Divide the numbers:

\frac{6}{2} = 3

= \frac{3 ( 2 + 3 )}{( 2 π - 3 )}

Remove parentheses:

(a) = a

= \frac{3 ( 2 + 3 )}{2 π - 3}

= \frac{3 ( 2 + 3 )}{2 π - 3}

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Frequently Asked Questions (FAQ)

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