What is the value of 2^{1/6}cos(pi/(12)) ?

The value of 2^{1/6}cos(pi/(12)) is (sqrt(3)+1)/(2*2^{1/3)}

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2^{1/6}cos(pi/(12))

Solution

2^{\frac{1}{6}} cos (\frac{π}{12})

Solution

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

Decimal Notation

1.08421 \dots

Solution steps

2^{\frac{1}{6}} cos (\frac{π}{12})

Rewrite using trig identities:

cos (\frac{π}{12}) = \frac{3 + 1}{2 \cdot 2 ^{\frac{1}{2}}}

cos (\frac{π}{12})

Rewrite using trig identities:

cos (\frac{π}{4}) cos (\frac{π}{6}) + sin (\frac{π}{4}) sin (\frac{π}{6})

cos (\frac{π}{12})

Write

cos (\frac{π}{12})

cos (\frac{π}{4} - \frac{π}{6})

= cos (\frac{π}{4} - \frac{π}{6})

Use the Angle Difference identity:

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

= cos (\frac{π}{4}) cos (\frac{π}{6}) + sin (\frac{π}{4}) sin (\frac{π}{6})

= cos (\frac{π}{4}) cos (\frac{π}{6}) + sin (\frac{π}{4}) sin (\frac{π}{6})

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:

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 (\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}

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}

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

Simplify

\frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} : \frac{3 + 1}{2 \cdot 2 ^{\frac{1}{2}}}

\frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2}

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

\frac{2}{2} \cdot \frac{3}{2}

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

\frac{2}{2}

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot 3 = 3

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

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

\frac{2}{2} \cdot \frac{1}{2}

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

\frac{2}{2}

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

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

Apply rule

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

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

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

= 2^{\frac{1}{6}} \cdot \frac{3 + 1}{2 \cdot 2 ^{\frac{1}{2}}}

Simplify

2^{\frac{1}{6}} \cdot \frac{3 + 1}{2 \cdot 2 ^{\frac{1}{2}}} : \frac{3 + 1}{2 \cdot 2 ^{\frac{1}{3}}}

2^{\frac{1}{6}} \cdot \frac{3 + 1}{2 \cdot 2 ^{\frac{1}{2}}}

Multiply fractions:

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

= \frac{( 3 + 1 ) \cdot 2 ^{\frac{1}{6}}}{2 \cdot 2 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{2 ^{\frac{1}{6}}}{2} = \frac{1}{2 ^{1 - \frac{1}{6}}}

= \frac{3 + 1}{2 ^{\frac{1}{2}} \cdot 2 ^{- \frac{1}{6} + 1}}

Subtract the numbers:

1 - \frac{1}{6} = \frac{5}{6}

= \frac{3 + 1}{2 ^{\frac{5}{6}} \cdot 2 ^{\frac{1}{2}}}

2^{\frac{5}{6}} \cdot 2^{\frac{1}{2}} = 2^{\frac{5}{6} + \frac{1}{2}}

2^{\frac{5}{6}} \cdot 2^{\frac{1}{2}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{5}{6}} \cdot 2^{\frac{1}{2}} = 2^{\frac{5}{6} + \frac{1}{2}}

= 2^{\frac{5}{6} + \frac{1}{2}}

= \frac{3 + 1}{2 ^{\frac{5}{6} + \frac{1}{2}}}

2^{\frac{5}{6} + \frac{1}{2}} = 2 \cdot 2^{\frac{1}{3}}

2^{\frac{5}{6} + \frac{1}{2}}

Join

\frac{5}{6} + \frac{1}{2} : \frac{4}{3}

\frac{5}{6} + \frac{1}{2}

Least Common Multiplier of

6, 2 : 6

6, 2

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

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

6

2

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{1}{2} :

multiply the denominator and numerator by

3

\frac{1}{2} = \frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}

= \frac{5}{6} + \frac{3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{5 + 3}{6}

Add the numbers:

5 + 3 = 8

= \frac{8}{6}

Cancel the common factor:

2

= \frac{4}{3}

= 2^{\frac{4}{3}}

2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}}

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

Apply exponent rule:

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

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

Refine

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

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

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

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Frequently Asked Questions (FAQ)

What is the value of 2^{1/6}cos(pi/(12)) ?
The value of 2^{1/6}cos(pi/(12)) is (sqrt(3)+1)/(2*2^{1/3)}