For this question, you could have estimated
[tex]\sin31^\circ\approx\sin30^\circ=\dfrac12[/tex]
so that you would expect about
[tex]\sin31^\circ\approx\dfrac12\approx\dfrac x7\implies x\approx3.5[/tex]
In general, computing [tex]\sin\theta[/tex] ([tex]\theta[/tex] in radians) by hand is tedious and difficult, but there are several ways to do it. Calculutors typically use well-known approximations in the form of truncated series expansions, such as
[tex]\sin\theta\approx\theta-\dfrac{\theta^3}6+\dfrac{\theta^5}{120}[/tex]
Then
[tex]\sin31^\circ=\sin\dfrac{31\pi}{180}\approx\sin\dfrac{31\cdot3.14}{180}\approx0.5148[/tex]
