What is the value of 25picos((pi 1/2)/2) ?

The value of 25picos((pi 1/2)/2) is (25sqrt(2)pi)/2

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25picos((pi 1/2)/2)

25 π cos (\frac{π \frac{1}{2}}{2})

\frac{25 2 π}{2}

Decimal

55.53603 \dots

Solution steps

25 π cos (\frac{π \frac{1}{2}}{2})

Simplify:

\frac{π \frac{1}{2}}{2} = \frac{π}{4}

\frac{π \frac{1}{2}}{2}

Multiply

π \frac{1}{2} : \frac{π}{2}

π \frac{1}{2}

Multiply fractions:

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

= \frac{1 π}{2}

Multiply:

1 π = π

= \frac{π}{2}

= \frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

= 25 π cos (\frac{π}{4})

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}

= 25 π \frac{2}{2}

Simplify

25 π \frac{2}{2} : \frac{25 2 π}{2}

25 π \frac{2}{2}

Multiply fractions:

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

= \frac{2 \cdot 25 π}{2}

Apply radical rule:

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

= \frac{25 \cdot 2 ^{\frac{1}{2}} π}{2}

Apply exponent rule:

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

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

= \frac{25 π}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{25 π}{2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{25 π}{2}

Rationalize

\frac{25 π}{2} : \frac{25 2 π}{2}

\frac{25 π}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{25 π 2}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{25 2 π}{2}

= \frac{25 2 π}{2}

= \frac{25 2 π}{2}

What is the value of 25picos((pi 1/2)/2) ?
The value of 25picos((pi 1/2)/2) is (25sqrt(2)pi)/2