sin 11 pie/12 + sin 7pie/12

Question

use the sum to product formula to find the exact value of 

Answer 1

Sum to Product formula for Sine function is

sin(u) + sin (v) = 2sin ((u+v)/2).cos ((u-v)/2)

 

Substitute u = 11.pi/12 and v = 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 3^rd Quadrant where Sine is positive

                                ....angle pi/6 = 30 degrees which is in 1^st Quadrant where cosine is positive

 

=2.(1/sqrt(2)).(sqrt(3)/2)

=sqrt(3/2)

= sqrt(1.5)

=1.225

 

Answer 2

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