What is the value of 4.5cos(pi/6 (12-1))+18.6 ?

The value of 4.5cos(pi/6 (12-1))+18.6 is (45sqrt(3)+372)/(20)

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4.5cos(pi/6 (12-1))+18.6

Solution

4.5 cos (\frac{π}{6} (12 - 1)) + 18.6

Solution

\frac{45 3 + 372}{20}

Decimal Notation

22.49711 \dots

Solution steps

4.5 cos (\frac{π}{6} (12 - 1)) + 18.6

= \frac{9}{2} cos (\frac{π}{6} (12 - 1)) + \frac{93}{5}

Simplify:

\frac{π}{6} (12 - 1) = \frac{11 π}{6}

\frac{π}{6} (12 - 1)

Multiply fractions:

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

= \frac{π ( 12 - 1 )}{6}

Subtract the numbers:

12 - 1 = 11

= \frac{11 π}{6}

\frac{9}{2} cos (\frac{11 π}{6}) + \frac{93}{5} = \frac{45 cos ( \frac{11 π}{6} ) + 186}{10}

\frac{9}{2} cos (\frac{11 π}{6}) + \frac{93}{5}

Multiply

\frac{9}{2} cos (\frac{11 π}{6}) : \frac{9 cos ( \frac{11 π}{6} )}{2}

\frac{9}{2} cos (\frac{11 π}{6})

Multiply fractions:

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

= \frac{9 cos ( \frac{11 π}{6} )}{2}

= \frac{9 cos ( \frac{11 π}{6} )}{2} + \frac{93}{5}

Least Common Multiplier of

2, 5 : 10

2, 5

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

Multiply each factor the greatest number of times it occurs in either

2

5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

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

10

For

\frac{9 cos ( \frac{11 π}{6} )}{2} :

multiply the denominator and numerator by

5

\frac{9 cos ( \frac{11 π}{6} )}{2} = \frac{9 cos ( \frac{11 π}{6} ) \cdot 5}{2 \cdot 5} = \frac{45 cos ( \frac{11 π}{6} )}{10}

For

\frac{93}{5} :

multiply the denominator and numerator by

2

\frac{93}{5} = \frac{93 \cdot 2}{5 \cdot 2} = \frac{186}{10}

= \frac{45 cos ( \frac{11 π}{6} )}{10} + \frac{186}{10}

Since the denominators are equal, combine the fractions:

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

= \frac{45 cos ( \frac{11 π}{6} ) + 186}{10}

= \frac{45 cos ( \frac{11 π}{6} ) + 186}{10}

Rewrite using trig identities:

cos (\frac{11 π}{6}) = \frac{3}{2}

cos (\frac{11 π}{6})

Rewrite using trig identities:

cos (π) cos (\frac{5 π}{6}) - sin (π) sin (\frac{5 π}{6})

cos (\frac{11 π}{6})

Write

cos (\frac{11 π}{6})

cos (π + \frac{5 π}{6})

= cos (π + \frac{5 π}{6})

Use the Angle Sum identity:

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

= cos (π) cos (\frac{5 π}{6}) - sin (π) sin (\frac{5 π}{6})

= cos (π) cos (\frac{5 π}{6}) - sin (π) sin (\frac{5 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{5 π}{6}) = - \frac{3}{2}

cos (\frac{5 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{5 π}{6}) = \frac{1}{2}

sin (\frac{5 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{45 \cdot \frac{3}{2} + 186}{10}

Simplify

\frac{45 \cdot \frac{3}{2} + 186}{10} : \frac{45 3 + 372}{20}

\frac{45 \cdot \frac{3}{2} + 186}{10}

Multiply

45 \cdot \frac{3}{2} : \frac{45 3}{2}

45 \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{3 \cdot 45}{2}

= \frac{\frac{45 3}{2} + 186}{10}

Join

\frac{3 \cdot 45}{2} + 186 : \frac{45 3 + 372}{2}

\frac{3 \cdot 45}{2} + 186

Convert element to fraction:

186 = \frac{186 \cdot 2}{2}

= \frac{3 \cdot 45}{2} + \frac{186 \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 45 + 186 \cdot 2}{2}

Multiply the numbers:

186 \cdot 2 = 372

= \frac{45 3 + 372}{2}

= \frac{\frac{45 3 + 372}{2}}{10}

Apply the fraction rule:

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

= \frac{3 \cdot 45 + 372}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{45 3 + 372}{20}

= \frac{45 3 + 372}{20}

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Frequently Asked Questions (FAQ)

What is the value of 4.5cos(pi/6 (12-1))+18.6 ?
The value of 4.5cos(pi/6 (12-1))+18.6 is (45sqrt(3)+372)/(20)