What is the value of 1/2 (7)(10)(cos(pi/4))(-pi/(45)) ?

The value of 1/2 (7)(10)(cos(pi/4))(-pi/(45)) is -(7sqrt(2)pi)/(18)

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1/2 (7)(10)(cos(pi/4))(-pi/(45))

Solution

\frac{1}{2} (7) (10) (cos (\frac{π}{4})) (- \frac{π}{45})

Solution

- \frac{7 2 π}{18}

Decimal Notation

- 1.72778 \dots

Solution steps

\frac{1}{2} (7) (10) (cos (\frac{π}{4})) (- \frac{π}{45})

= \frac{1}{2} \cdot 7 \cdot 10 cos (\frac{π}{4}) (- \frac{π}{45})

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}

= \frac{1}{2} \cdot 7 \cdot 10 \cdot \frac{2}{2} (- \frac{π}{45})

Simplify

\frac{1}{2} \cdot 7 \cdot 10 \cdot \frac{2}{2} (- \frac{π}{45}) : - \frac{7 2 π}{18}

\frac{1}{2} \cdot 7 \cdot 10 \cdot \frac{2}{2} (- \frac{π}{45})

Remove parentheses:

(- a) = - a

= - \frac{1}{2} \cdot 7 \cdot 10 \cdot \frac{2}{2} \cdot \frac{π}{45}

Multiply fractions:

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

= - \frac{1 \cdot 2 π 7 \cdot 10}{2 \cdot 2 \cdot 45}

Multiply the numbers:

1 \cdot 7 \cdot 10 = 70

= - \frac{70 2 π}{2 \cdot 2 \cdot 45}

Multiply the numbers:

2 \cdot 2 \cdot 45 = 180

= - \frac{70 2 π}{180}

Cancel

\frac{70 2 π}{180} : \frac{7 π}{3 ^{2} 2}

\frac{70 2 π}{180}

Cancel the common factor:

10

= \frac{7 2 π}{18}

Factor

18 : 3^{2} \cdot 2

Factor

18 = 3^{2} \cdot 2

= \frac{7 2 π}{3 ^{2} \cdot 2}

Cancel

\frac{7 2 π}{2 \cdot 3 ^{2}} : \frac{7 π}{2 \cdot 3 ^{2}}

\frac{7 2 π}{2 \cdot 3 ^{2}}

Apply radical rule:

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

= \frac{7 \cdot 2 ^{\frac{1}{2}} π}{3 ^{2} \cdot 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{7 π}{3 ^{2} \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{7 π}{3 ^{2} \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{7 π}{3 ^{2} 2}

= \frac{7 π}{2 \cdot 3 ^{2}}

= - \frac{7 π}{3 ^{2} 2}

3^{2} = 9

= - \frac{7 π}{9 2}

Rationalize

- \frac{7 π}{9 2} : - \frac{7 2 π}{18}

- \frac{7 π}{9 2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{7 π 2}{2 \cdot 9 2}

2 \cdot 92 = 18

2 \cdot 92

Apply radical rule:

a a = a

22 = 2

= 9 \cdot 2

Multiply the numbers:

9 \cdot 2 = 18

= 18

= - \frac{7 2 π}{18}

= - \frac{7 2 π}{18}

= - \frac{7 2 π}{18}

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Frequently Asked Questions (FAQ)

What is the value of 1/2 (7)(10)(cos(pi/4))(-pi/(45)) ?
The value of 1/2 (7)(10)(cos(pi/4))(-pi/(45)) is -(7sqrt(2)pi)/(18)