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sin 11 pie/12 + sin 7pie/12

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2 Answers

Sum to Product formula for Sine function is
sin(u) + sin (v) = 2sin ((u+v)/2).cos ((u-v)/2)
 
Substitute u = http://11.pi/12 and v = http://7.pi/12 in above equation.
sin 11pi/12 + sin 7pi/12
=2sin((11pi/12+7pi/12)/2).cos((11pi/12-7pi/12)/2)
=2sin((18pi/12)/2).cos((4pi/12)/2)
=2sin((3pi/2)/2).cos((pi/3)/2)
=2sin(3pi/4).cos(pi/6) ....angle 3pi/4=135 degrees which is in 3rd Quadrant where Sine is positive
                                ....angle pi/6 = 30 degrees which is in 1st Quadrant where cosine is positive
 
=2.(1/sqrt(2)).(sqrt(3)/2)
=sqrt(3/2)
= sqrt(1.5)
=1.225
 
Sin 11∏/12 + Sin 7∏/ 12 =
 
Sin ( ∏/ 12)  + Sin ( 6∏/12 + ∏/12) =
 
Sin ( ∏/4 - ∏/6) + Sin( ∏/4 + ∏/ 6)=
 
Sin ∏/4 . Cos ∏/6 - Sin ∏/6 cos ∏/4 + Sin ∏/4 cos ∏/6 + Sin∏/6 cos ∏/4 = 
 
 2 Sin ∏/4 . Cos ∏/6 =
 
 2 √2 / 2 . √3 /2 = √6 / 2