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

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

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(pi/4 cos(pi/4)-sin(pi/4))/((pi/4)^2)

Solution

\frac{\frac{π}{4} cos ( \frac{π}{4} ) - sin ( \frac{π}{4} )}{( \frac{π}{4} ) ^{2}}

Solution

\frac{2 2 ( π - 4 )}{π ^{2}}

Decimal Notation

- 0.24600 \dots

Solution steps

\frac{\frac{π}{4} cos ( \frac{π}{4} ) - sin ( \frac{π}{4} )}{( \frac{π}{4} ) ^{2}}

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

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{2}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

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{2}{2}

= \frac{\frac{π}{4} \cdot \frac{2}{2} - \frac{2}{2}}{( \frac{π}{4} ) ^{2}}

Simplify

\frac{\frac{π}{4} \cdot \frac{2}{2} - \frac{2}{2}}{( \frac{π}{4} ) ^{2}} : \frac{2 2 ( π - 4 )}{π ^{2}}

\frac{\frac{π}{4} \cdot \frac{2}{2} - \frac{2}{2}}{( \frac{π}{4} ) ^{2}}

(\frac{π}{4})^{2} = \frac{π ^{2}}{4 ^{2}}

(\frac{π}{4})^{2}

Apply exponent rule:

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

= \frac{π ^{2}}{4 ^{2}}

= \frac{\frac{π}{4} \cdot \frac{2}{2} - \frac{2}{2}}{\frac{π ^{2}}{4 ^{2}}}

\frac{π}{4} \cdot \frac{2}{2} = \frac{2 π}{8}

\frac{π}{4} \cdot \frac{2}{2}

Multiply fractions:

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

= \frac{π 2}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{2 π}{8}

= \frac{\frac{2 π}{8} - \frac{2}{2}}{\frac{π ^{2}}{4 ^{2}}}

Apply the fraction rule:

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

= \frac{( \frac{π 2}{8} - \frac{2}{2} ) \cdot 4 ^{2}}{π ^{2}}

Join

\frac{π 2}{8} - \frac{2}{2} : \frac{π - 4}{4 2}

\frac{π 2}{8} - \frac{2}{2}

Least Common Multiplier of

8, 2 : 8

8, 2

Least Common Multiplier (LCM)

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

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

8

2

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

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

8

For

\frac{2}{2} :

multiply the denominator and numerator by

4

\frac{2}{2} = \frac{2 \cdot 4}{2 \cdot 4} = \frac{2 \cdot 4}{8}

= \frac{π 2}{8} - \frac{2 \cdot 4}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - 2 \cdot 4}{8}

Factor out common term

2

= \frac{2 ( π - 4 )}{8}

Factor

8 : 2^{3}

Factor

8 = 2^{3}

= \frac{2 ( π - 4 )}{2 ^{3}}

Cancel

\frac{2 ( π - 4 )}{2 ^{3}} : \frac{π - 4}{2 ^{\frac{5}{2}}}

\frac{2 ( π - 4 )}{2 ^{3}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{π - 4}{2 ^{3 - \frac{1}{2}}}

Subtract the numbers:

3 - \frac{1}{2} = \frac{5}{2}

= \frac{π - 4}{2 ^{\frac{5}{2}}}

= \frac{π - 4}{2 ^{\frac{5}{2}}}

2^{\frac{5}{2}} = 2^{2} 2

2^{\frac{5}{2}}

2^{\frac{5}{2}} = 2^{2 + \frac{1}{2}}

= 2^{2 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 2^{2} 2

= \frac{π - 4}{2 ^{2} 2}

2^{2} = 4

= \frac{π - 4}{4 2}

= \frac{4 ^{2} \cdot \frac{π - 4}{4 2}}{π ^{2}}

4^{2} = 16

= \frac{16 \cdot \frac{π - 4}{4 2}}{π ^{2}}

Multiply

\frac{π - 4}{4 2} \cdot 16 : 22 (π - 4)

\frac{π - 4}{4 2} \cdot 16

Multiply fractions:

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

= \frac{( π - 4 ) \cdot 16}{4 2}

Divide the numbers:

\frac{16}{4} = 4

= \frac{4 ( π - 4 )}{2}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 ^{2} ( π - 4 )}{2}

Cancel

\frac{2 ^{2} ( π - 4 )}{2} : 2^{\frac{3}{2}} (π - 4)

\frac{2 ^{2} ( π - 4 )}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{2} ( π - 4 )}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{2}}{2 ^{\frac{1}{2}}} = 2^{2 - \frac{1}{2}}

= 2^{- \frac{1}{2} + 2} (π - 4)

Subtract the numbers:

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

= 2^{\frac{3}{2}} (π - 4)

= 2^{\frac{3}{2}} (π - 4)

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

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

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 22

= 22 (π - 4)

= \frac{2 2 ( π - 4 )}{π ^{2}}

= \frac{2 2 ( π - 4 )}{π ^{2}}

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Frequently Asked Questions (FAQ)

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