Find $\sin 31°$ using approximation (given $\sin 30° = 0.5$, $\cos 30° = \sqrt 3/2 \approx 0.866$, $1° = 0.01745$ rad):

$\sin(x + dx) \approx \sin x + \cos x \cdot dx$. $\sin 31° = \sin(30° + 1°) \approx \sin 30° + \cos 30° \cdot (1° \text{ in rad}) = 0.5 + 0.866 \times 0.01745 = 0.5 + 0.01511 \approx 0.5151$.