b. In triangle \( A B C \), with side of length \( a \) opposite angle \( A \) etc, find \( a \) if \( A=0.9 \), \( B=0.7 \) and \( b=50.5 \mathrm{~cm} \).

To find the length of side \( a \) in triangle \( ABC \) using the given angles \( A \) and \( B \), and side \( b \), we can use the Law of Sines. The Law of Sines states that: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] Given: - \( A = 0.9 \) radians - \( B = 0.7 \) radians - \( b = 50.5 \, \text{cm} \) First, we need to find angle \( C \): \[ C = \pi - A - B = \pi - 0.9 - 0.7 = \pi - 1.6 \approx 1.5416 \, \text{radians} \] Now we can use the Law of Sines to find \( a \): \[ \frac{a}{\sin A} = \frac{b}{\sin B} \] Substituting the known values: \[ \frac{a}{\sin(0.9)} = \frac{50.5}{\sin(0.7)} \] Now we calculate \( \sin(0.9) \) and \( \sin(0.7) \): \[ \sin(0.9) \approx 0.6216 \] \[ \sin(0.7) \approx 0.6442 \] Now substituting these values into the equation: \[ \frac{a}{0.6216} = \frac{50.5}{0.6442} \] Cross-multiplying gives: \[ a \cdot 0.6442 = 50.5 \cdot 0.6216 \] Calculating the right side: \[ 50.5 \cdot 0.6216 \approx 31.3968 \] Now we can solve for \( a \): \[ a = \frac{31.3968}{0.6442} \approx 48.8 \, \text{cm} \] Thus, the length of side \( a \) is approximately: \[ \boxed{48.8 \, \text{cm}} \]