Expert: Ahmed Salami - 3/16/2009

Question
I have a problem that says : express sin11pi/12+sin5pi/12 as a product and find the exact value of the product I have no clue aout this problem I was wondering could you work this out and show me how u did it please

Answer
Hi Jamal,
First, you need to remember the double angles identity;
sin(A+B) = sinA.cosB + sinB.cosA
sin(A-B) = sinA.cosB - sinB.cosA
Now, adding the two equations, we have
sin(A+B) + sin(A-B) = 2sinA.cosB
In our problem, we have
sin(11pi/12) + sin(5pi/12)
and in order to relate this to the identity we need vaues of A and B such that
A + B = 11pi/12
A - B = 5pi/12
The values are (and i'm sure you can figure this out)
A = 8pi/12
 = 2pi/3
and
B = 3pi/12
 = pi/4
Therefore,
sin(11pi/12) + sin(5pi/12) = 2sin(2pi/3).cos(pi/4)
                          = 2[(sqrt3)/2].[1/(sqrt2)]
                          = sqrt3/sqrt2
                          = sqrt(3/2)

Hope it helps.

Regards