How to evaluate 3pi/8 using the half angle identity for sin^2 theta?

Question

I am looking for an exact answer, not a decimal. 

Answer 1

 

Note the half angle identity

 

     cos(2θ) = 1 - 2sin²θ

 

which is equivalent of

 

     sin²θ = (1 - cos(2θ))/2

 

Putting 3π/8 into θ in the equation right above

 

     sin²(3π/8) = (1 - cos(2·3π/8))/2

 

                     = (1 - cos(3π/4))/2

 

                     = (1 - (-√2/2))/2        (where, cos(3π/4) = -√2/2)

 

                     = (2 + √2)/4

 

As 3π/8 is the angle in the first quadrant,

 

sin(3π/8) must be positive. So,

 

     sin(3π/8) = √((2 + √2)/4)

 

                    = (1/2)√(2 + √2)