What is the value of (pi/6 cos(pi/6)-sin(pi/6))/((pi/6)^2) ?

The value of (pi/6 cos(pi/6)-sin(pi/6))/((pi/6)^2) is (3(sqrt(3)pi-6))/(pi^2)

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(pi/6 cos(pi/6)-sin(pi/6))/((pi/6)^2)

Solution

\frac{\frac{π}{6} cos ( \frac{π}{6} ) - sin ( \frac{π}{6} )}{( \frac{π}{6} ) ^{2}}

Solution

\frac{3 ( 3 π - 6 )}{π ^{2}}

Decimal Notation

- 0.16979 \dots

Solution steps

\frac{\frac{π}{6} cos ( \frac{π}{6} ) - sin ( \frac{π}{6} )}{( \frac{π}{6} ) ^{2}}

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{π}{6})

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{3}{2}

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{π}{6})

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{1}{2}

= \frac{\frac{π}{6} \cdot \frac{3}{2} - \frac{1}{2}}{( \frac{π}{6} ) ^{2}}

Simplify

\frac{\frac{π}{6} \cdot \frac{3}{2} - \frac{1}{2}}{( \frac{π}{6} ) ^{2}} : \frac{3 ( 3 π - 6 )}{π ^{2}}

\frac{\frac{π}{6} \cdot \frac{3}{2} - \frac{1}{2}}{( \frac{π}{6} ) ^{2}}

(\frac{π}{6})^{2} = \frac{π ^{2}}{6 ^{2}}

(\frac{π}{6})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{π ^{2}}{6 ^{2}}

= \frac{\frac{π}{6} \cdot \frac{3}{2} - \frac{1}{2}}{\frac{π ^{2}}{6 ^{2}}}

\frac{π}{6} \cdot \frac{3}{2} = \frac{3 π}{12}

\frac{π}{6} \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{π 3}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{3 π}{12}

= \frac{\frac{3 π}{12} - \frac{1}{2}}{\frac{π ^{2}}{6 ^{2}}}

Apply the fraction rule:

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

= \frac{( \frac{π 3}{12} - \frac{1}{2} ) \cdot 6 ^{2}}{π ^{2}}

Join

\frac{π 3}{12} - \frac{1}{2} : \frac{3 π - 6}{12}

\frac{π 3}{12} - \frac{1}{2}

Least Common Multiplier of

12, 2 : 12

12, 2

Least Common Multiplier (LCM)

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 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

12

2

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{1}{2} :

multiply the denominator and numerator by

6

\frac{1}{2} = \frac{1 \cdot 6}{2 \cdot 6} = \frac{6}{12}

= \frac{π 3}{12} - \frac{6}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - 6}{12}

= \frac{6 ^{2} \cdot \frac{3 π - 6}{12}}{π ^{2}}

6^{2} = 36

= \frac{36 \cdot \frac{3 π - 6}{12}}{π ^{2}}

Multiply

\frac{π 3 - 6}{12} \cdot 36 : 3 (3 π - 6)

\frac{π 3 - 6}{12} \cdot 36

Multiply fractions:

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

= \frac{( π 3 - 6 ) \cdot 36}{12}

Divide the numbers:

\frac{36}{12} = 3

= 3 (3 π - 6)

= \frac{3 ( 3 π - 6 )}{π ^{2}}

= \frac{3 ( 3 π - 6 )}{π ^{2}}

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Frequently Asked Questions (FAQ)

What is the value of (pi/6 cos(pi/6)-sin(pi/6))/((pi/6)^2) ?
The value of (pi/6 cos(pi/6)-sin(pi/6))/((pi/6)^2) is (3(sqrt(3)pi-6))/(pi^2)