Question

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

Hint:

To approximate trigonometric values for small angles, convert degrees to radians and use known values or approximations.

Answer 1

Step 1: Convert \( 30^\circ 30' \) to decimal degrees: \[ 30^\circ 30' = 30 + \frac{30}{60} = 30.5^\circ \] Step 2: Convert \( 30.5^\circ \) to radians: \[ 30.5^\circ = 30.5 \times 0.0175 = 0.53375 \text{ radians} \] Step 3: Use the approximation for \( \sin{x} \) near \( x = 0 \), or directly calculate using a calculator: \[ \sin{30.5^\circ} \approx 0.500 \] Thus, the approximate value of \( \sin{30^\circ 30'} \) is: \[ \boxed{0.500} \]