What is the value of cos((11pi}{12})cos((3pi)/4)+sin((11pi)/(12))sin(\frac{3pi)/4) ?

The value of cos((11pi}{12})cos((3pi)/4)+sin((11pi)/(12))sin(\frac{3pi)/4) is (sqrt(3))/2

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Popular >

cos((11pi}{12})cos((3pi)/4)+sin((11pi)/(12))sin(\frac{3pi)/4)

Solution

cos (\frac{11 π}{12}) cos (\frac{3 π}{4}) + sin (\frac{11 π}{12}) sin (\frac{3 π}{4})

Solution

\frac{3}{2}

Decimal Notation

0.86602 \dots

Solution steps

cos (\frac{11 π}{12}) cos (\frac{3 π}{4}) + sin (\frac{11 π}{12}) sin (\frac{3 π}{4})

Rewrite using trig identities:

cos (\frac{π}{6})

cos (\frac{11 π}{12}) cos (\frac{3 π}{4}) + sin (\frac{11 π}{12}) sin (\frac{3 π}{4})

Use the Angle Difference identity:

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

= cos (\frac{11 π}{12} - \frac{3 π}{4})

Simplify:

\frac{11 π}{12} - \frac{3 π}{4} = \frac{π}{6}

\frac{11 π}{12} - \frac{3 π}{4}

Least Common Multiplier of

12, 4 : 12

12, 4

Least Common Multiplier (LCM)

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

12

4

= 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{3 π}{4} :

multiply the denominator and numerator by

3

\frac{3 π}{4} = \frac{3 π 3}{4 \cdot 3} = \frac{9 π}{12}

= \frac{11 π}{12} - \frac{9 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{11 π - 9 π}{12}

Add similar elements:

11 π - 9 π = 2 π

= \frac{2 π}{12}

Cancel the common factor:

2

= \frac{π}{6}

= cos (\frac{π}{6})

= cos (\frac{π}{6})

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}

= \frac{3}{2}

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Frequently Asked Questions (FAQ)

What is the value of cos((11pi}{12})cos((3pi)/4)+sin((11pi)/(12))sin(\frac{3pi)/4) ?
The value of cos((11pi}{12})cos((3pi)/4)+sin((11pi)/(12))sin(\frac{3pi)/4) is (sqrt(3))/2