#Python 2, 100 bytes

from math import*
x,a,b=input()
while a<=b:
    if round(sin(pi*a/180),5)==x:print a
    a=round(a+.001,3)

Try it online

Python 2, 100 bytes

from math import*
x,a,b=input()
while a<=b:
    if round(sin(pi*a/180),5)==x:print a
    a=round(a+.001,3)

Try it online

#Python 2, 100 bytes

The question currently has incorrect examples (see my comments), so I went with the inequality that includes a and b, rather than the example which leaves out those solutions.

from math import*
x,a,b=input()
while a<=b:
    if round(sin(pi*a/180),5)==x:print a
    a=round(a+.001,3)

Try it online

#Python 2, 100 bytes

from math import*
x,a,b=input()
while a<=b:
    if round(sin(pi*a/180),5)==x:print a
    a=round(a+.001,3)

Try it online