What is the value of cos((7pi)/(12))cos((5pi)/(12))+sin((7pi)/(12))sin((5pi)/(12)) ?

The value of cos((7pi)/(12))cos((5pi)/(12))+sin((7pi)/(12))sin((5pi)/(12)) is (sqrt(3))/2

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cos((7pi)/(12))cos((5pi)/(12))+sin((7pi)/(12))sin((5pi)/(12))

cos (\frac{7 π}{12}) cos (\frac{5 π}{12}) + sin (\frac{7 π}{12}) sin (\frac{5 π}{12})

\frac{3}{2}

Decimal Notation

0.86602 \dots

Solution steps

cos (\frac{7 π}{12}) cos (\frac{5 π}{12}) + sin (\frac{7 π}{12}) sin (\frac{5 π}{12})

Rewrite using trig identities:

cos (\frac{π}{6})

cos (\frac{7 π}{12}) cos (\frac{5 π}{12}) + sin (\frac{7 π}{12}) sin (\frac{5 π}{12})

Use the Angle Difference identity:

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

= cos (\frac{7 π}{12} - \frac{5 π}{12})

Simplify:

\frac{7 π}{12} - \frac{5 π}{12} = \frac{π}{6}

\frac{7 π}{12} - \frac{5 π}{12}

Apply rule

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

= \frac{7 π - 5 π}{12}

Add similar elements:

7 π - 5 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}

cot (\frac{- π}{4})

cos (90 0^{\circ})

sin (8 3^{\circ})

sin (2) (60)

\frac{sin ( π )}{6}

What is the value of cos((7pi)/(12))cos((5pi)/(12))+sin((7pi)/(12))sin((5pi)/(12)) ?
The value of cos((7pi)/(12))cos((5pi)/(12))+sin((7pi)/(12))sin((5pi)/(12)) is (sqrt(3))/2