(4) Calculate the values of angles \( A \) and \( C \) of \( \triangle A B C \), where \( b=14.35 \mathrm{~cm}, a=7.82 \mathrm{~cm} \) and \( B=115.6^{\circ} \).(WAEC)

To find the values of angles \( A \) and \( C \) in triangle \( ABC \), we can use the Law of Sines, which states: \[ \frac{a}{\sin A} = \frac{b}{\sin B} \] Given \( a = 7.82 \, \text{cm} \), \( b = 14.35 \, \text{cm} \), and \( B = 115.6^\circ \): 1. Rearrange the equation to solve for \( A \): \[ \sin A = \frac{a \cdot \sin B}{b} \] Substituting the values: \[ \sin A = \frac{7.82 \cdot \sin(115.6^\circ)}{14.35} \] Calculating \( \sin(115.6^\circ) \): \[ \sin(115.6^\circ) \approx 0.907 \] Now substituting: \[ \sin A = \frac{7.82 \cdot 0.907}{14.35} \approx 0.503 \] Now find \( A \): \[ A \approx \arcsin(0.503) \approx 30.2^\circ \] 2. Now, to find angle \( C \): Using the angle sum property of triangles: \[ C = 180^\circ - A - B = 180^\circ - 30.2^\circ - 115.6^\circ \approx 34.2^\circ \] Thus, the values for angles \( A \) and \( C \) are approximately \( 30.2^\circ \) and \( 34.2^\circ \) respectively.