What is the value of cos((5pi)/4-pi/6) ?

The value of cos((5pi)/4-pi/6) is (sqrt(2)(-sqrt(3)-1))/4

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cos((5pi)/4-pi/6)

Solution

cos (\frac{5 π}{4} - \frac{π}{6})

Solution

\frac{2 ( - 3 - 1 )}{4}

Decimal Notation

- 0.96592 \dots

Solution steps

cos (\frac{5 π}{4} - \frac{π}{6})

Simplify:

\frac{5 π}{4} - \frac{π}{6} = \frac{13 π}{12}

\frac{5 π}{4} - \frac{π}{6}

Least Common Multiplier of

4, 6 : 12

4, 6

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

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

4

6

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{5 π}{4} :

multiply the denominator and numerator by

3

\frac{5 π}{4} = \frac{5 π 3}{4 \cdot 3} = \frac{15 π}{12}

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

= \frac{15 π}{12} - \frac{π 2}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{15 π - π 2}{12}

Add similar elements:

15 π - 2 π = 13 π

= \frac{13 π}{12}

= cos (\frac{13 π}{12})

Rewrite using trig identities:

cos (\frac{5 π}{6}) cos (\frac{π}{4}) - sin (\frac{5 π}{6}) sin (\frac{π}{4})

cos (\frac{13 π}{12})

Write

cos (\frac{13 π}{12})

cos (\frac{5 π}{6} + \frac{π}{4})

= cos (\frac{5 π}{6} + \frac{π}{4})

Use the Angle Sum identity:

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

= cos (\frac{5 π}{6}) cos (\frac{π}{4}) - sin (\frac{5 π}{6}) sin (\frac{π}{4})

= cos (\frac{5 π}{6}) cos (\frac{π}{4}) - sin (\frac{5 π}{6}) sin (\frac{π}{4})

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:

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:

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}

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}

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

Simplify

(- \frac{3}{2}) \frac{2}{2} - \frac{1}{2} \cdot \frac{2}{2} : \frac{2 ( - 3 - 1 )}{4}

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

Remove parentheses:

(- a) = - a

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

Factor out common term

\frac{2}{2}

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

- \frac{3}{2} - \frac{1}{2} = \frac{- 3 - 1}{2}

- \frac{3}{2} - \frac{1}{2}

Apply rule

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

= \frac{- 3 - 1}{2}

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

Multiply fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 ( - 1 - 3 )}{4}

= \frac{2 ( - 3 - 1 )}{4}

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Frequently Asked Questions (FAQ)

What is the value of cos((5pi)/4-pi/6) ?
The value of cos((5pi)/4-pi/6) is (sqrt(2)(-sqrt(3)-1))/4