Use a halfangle formula to find the exact value of sin 5 pi8

Use a half-angle formula to find the exact value of sin 5 pi/8.

Solution

sin(5pi/8)

Now use half angle formula :

sin(x/2) = sqrt[ (1+cosx)/2]

Now we need value of cos(5pi*2/8) = cos5pi/4 to plug in formula

So, cos5pi/4 = cos(pi +pi/4) = - cospi/4 = - 1/sqrt2

So, sin( 5pi/8) = sqr[( 1+ cos5pi/4)/2]

= sqrt[ (1- 1/sqrt2)/2]

= sqrt(sqrt2 -1)/2