What is the value of cos((11pi)/(12))cos((7pi)/(12))+sin((11pi)/(12))sin((7pi)/(12)) ?

The value of cos((11pi)/(12))cos((7pi)/(12))+sin((11pi)/(12))sin((7pi)/(12)) is 1/2

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cos((11pi)/(12))cos((7pi)/(12))+sin((11pi)/(12))sin((7pi)/(12))

cos (\frac{11 π}{12}) cos (\frac{7 π}{12}) + sin (\frac{11 π}{12}) sin (\frac{7 π}{12})

\frac{1}{2}

Decimal Notation

0.5

Solution steps

cos (\frac{11 π}{12}) cos (\frac{7 π}{12}) + sin (\frac{11 π}{12}) sin (\frac{7 π}{12})

Rewrite using trig identities:

cos (\frac{π}{3})

cos (\frac{7 π}{12}) cos (\frac{11 π}{12}) + sin (\frac{7 π}{12}) sin (\frac{11 π}{12})

Use the Angle Difference identity:

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

= cos (\frac{7 π}{12} - \frac{11 π}{12})

Simplify:

\frac{7 π}{12} - \frac{11 π}{12} = - \frac{π}{3}

\frac{7 π}{12} - \frac{11 π}{12}

Apply rule

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

= \frac{7 π - 11 π}{12}

Add similar elements:

7 π - 11 π = - 4 π

= \frac{- 4 π}{12}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4 π}{12}

Cancel the common factor:

4

= - \frac{π}{3}

= cos (- \frac{π}{3})

Use the following property:

cos (- x) = cos (x)

cos (- \frac{π}{3}) = cos (\frac{π}{3})

= cos (\frac{π}{3})

= cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

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{1}{2}

= \frac{1}{2}

sec (57 0^{\circ})

90 sin (3 0^{\circ})

sec^{2} (0) - 1

1 sin (\frac{π}{4})

sin (\frac{10 π}{2})

What is the value of cos((11pi)/(12))cos((7pi)/(12))+sin((11pi)/(12))sin((7pi)/(12)) ?
The value of cos((11pi)/(12))cos((7pi)/(12))+sin((11pi)/(12))sin((7pi)/(12)) is 1/2