What is the value of 3.05cos(pi/6 7)+19.15 ?

The value of 3.05cos(pi/6 7)+19.15 is (-61sqrt(3)+766)/(40)

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3.05cos(pi/6 7)+19.15

Solution

3.05 cos (\frac{π}{6} 7) + 19.15

Solution

\frac{- 61 3 + 766}{40}

Decimal Notation

16.50862 \dots

Solution steps

3.05 cos (\frac{π}{6} \cdot 7) + 19.15

= \frac{61}{20} cos (\frac{π}{6} \cdot 7) + \frac{383}{20}

Simplify:

\frac{π}{6} \cdot 7 = \frac{7 π}{6}

\frac{π}{6} \cdot 7

Multiply fractions:

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

= \frac{π 7}{6}

\frac{61}{20} cos (\frac{7 π}{6}) + \frac{383}{20} = \frac{61 cos ( \frac{7 π}{6} ) + 383}{20}

\frac{61}{20} cos (\frac{7 π}{6}) + \frac{383}{20}

Multiply

\frac{61}{20} cos (\frac{7 π}{6}) : \frac{61 cos ( \frac{7 π}{6} )}{20}

\frac{61}{20} cos (\frac{7 π}{6})

Multiply fractions:

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

= \frac{61 cos ( \frac{7 π}{6} )}{20}

= \frac{61 cos ( \frac{7 π}{6} )}{20} + \frac{383}{20}

Apply rule

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

= \frac{61 cos ( \frac{7 π}{6} ) + 383}{20}

= \frac{61 cos ( \frac{7 π}{6} ) + 383}{20}

Rewrite using trig identities:

cos (\frac{7 π}{6}) = - \frac{3}{2}

cos (\frac{7 π}{6})

Rewrite using trig identities:

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

cos (\frac{7 π}{6})

Write

cos (\frac{7 π}{6})

cos (π + \frac{π}{6})

= cos (π + \frac{π}{6})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (π) cos (\frac{π}{6}) - sin (π) sin (\frac{π}{6})

= cos (π) cos (\frac{π}{6}) - sin (π) sin (\frac{π}{6})

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

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 (π) = 0

sin (π)

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}

= 0

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}

= (- 1) \frac{3}{2} - 0 \cdot \frac{1}{2}

Simplify

= - \frac{3}{2}

= \frac{61 ( - \frac{3}{2} ) + 383}{20}

Simplify

\frac{61 ( - \frac{3}{2} ) + 383}{20} : \frac{- 61 3 + 766}{40}

\frac{61 ( - \frac{3}{2} ) + 383}{20}

Remove parentheses:

(- a) = - a

= \frac{- 61 \cdot \frac{3}{2} + 383}{20}

Multiply

61 \cdot \frac{3}{2} : \frac{61 3}{2}

61 \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{3 \cdot 61}{2}

= \frac{- \frac{61 3}{2} + 383}{20}

Join

- \frac{3 \cdot 61}{2} + 383 : \frac{- 61 3 + 766}{2}

- \frac{3 \cdot 61}{2} + 383

Convert element to fraction:

383 = \frac{383 \cdot 2}{2}

= - \frac{3 \cdot 61}{2} + \frac{383 \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{- 3 \cdot 61 + 383 \cdot 2}{2}

Multiply the numbers:

383 \cdot 2 = 766

= \frac{- 61 3 + 766}{2}

= \frac{\frac{- 61 3 + 766}{2}}{20}

Apply the fraction rule:

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

= \frac{- 3 \cdot 61 + 766}{2 \cdot 20}

Multiply the numbers:

2 \cdot 20 = 40

= \frac{- 61 3 + 766}{40}

= \frac{- 61 3 + 766}{40}

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Frequently Asked Questions (FAQ)

What is the value of 3.05cos(pi/6 7)+19.15 ?
The value of 3.05cos(pi/6 7)+19.15 is (-61sqrt(3)+766)/(40)