Question

Find the approximate value of \( \sin (30^\circ 30') \). Given that \( 1^\circ = 0.0175^c \) and \( \cos 30^\circ = 0.866 \).

Hint:

Use linear approximation for small angle changes; convert degrees to radians accurately.

Answer 1

Convert \( 30^\circ 30' = 30.5^\circ \). Since \( 1^\circ = 0.0175 \) radians, \( 30.5^\circ = 30.5 \cdot 0.0175 = 0.53375 \) radians. Use linear approximation: \[ \sin x \approx \sin x_0 + (x - x_0) \cos x_0, \quad x_0 = 30^\circ = \frac{\pi}{6} \approx 0.5236 \text{ radians}, \quad x = 0.53375. \] \[ \sin 30^\circ = \frac{1}{2} = 0.5, \quad \cos 30^\circ = 0.866, \quad x - x_0 = 0.53375 - 0.5236 = 0.01015. \] \[ \sin 30.5^\circ \approx 0.5 + 0.01015 \cdot 0.866 \approx 0.5 + 0.00879 = 0.50879. \] Answer: \( \sin 30.5^\circ \approx 0.509 \).