Using the sum-to-product formula for the sum of four sines:

sin5a + sin6a + sin7a + sin8a
= 2sin((5a + 8a)/2)cos((5a - 8a)/2) + 2sin((6a + 7a)/2)cos((6a - 7a)2)
= 2sin(6.5a)cos(-1/2a) + 2sin(6.5a)cos(1/2a)
= 2sin(6.5a)(cos(-1/2a) + cos(1/2a))
= 2sin(6.5a)(2cos(0)cos(1/2a))
= 4sin(6.5a)cos(1/2a)

Therefore, sin5a + sin6a + sin7a + sin8a= 4sin(6.5a)cos(1/2a).