The exact value of sin 5π / 12 is

5π/12 = π/6 + π/4 (another way to write it)

Use the angle addition formula for sine:
sin (a + b) = sin a * cosb + cos a * sin b

Substituting in a = π/6 and b = π/4
sin (π/6 + π/4) =
sin π/6 * cos π/4 + sin π/4 * cos π/6

There are exact values for all of the above:
sin π/6 = 1/2
cos π/4 = √2/2
sin π/4 = √2/2
cos π/6 = √3/2

Substituting these in:
1/2 * √2/2 + √2/2 * √3/2
= √2/4 + √6/4
= (√6+√2)/4

sin(5π/12)

= sin(π/4 + π/6)

= sin(π/4)cos(π/6) + cos(π/4)sin(π/6)

= (1/√2)(√3/2) + ( 1/√2)(1/2)

= (√3 + 1 )/2√2

= 1/4(√6 +√2 )

sin(5π/12) = (1 + √3)/(2√2) ≈ .9659

0.966