a. In triangle \( A B C \), with side of length \( a \) opposite angle \( A \) etc, find \( a \) if \( A=1 \), \( B=0.3 \) and \( b=34.77 \mathrm{~cm} \).

To find the length of side \( a \) in triangle \( ABC \) given angles \( A \) and \( B \) and side \( b \), we can use the Law of Sines, which states: \[ \frac{a}{\sin A} = \frac{b}{\sin B} \] We are given: - \( A = 1 \) radian - \( B = 0.3 \) radians - \( b = 34.77 \) cm First, we need to calculate \( \sin A \) and \( \sin B \): \[ \sin A = \sin(1) \quad \text{and} \quad \sin B = \sin(0.3) \] Using a calculator, we find: \[ \sin(1) \approx 0.8415 \] \[ \sin(0.3) \approx 0.2955 \] Now we can substitute these values into the Law of Sines equation: \[ \frac{a}{\sin(1)} = \frac{34.77}{\sin(0.3)} \] Substituting the sine values: \[ \frac{a}{0.8415} = \frac{34.77}{0.2955} \] Now, calculate the right side: \[ \frac{34.77}{0.2955} \approx 117.5 \] Now we have: \[ \frac{a}{0.8415} = 117.5 \] To find \( a \), multiply both sides by \( 0.8415 \): \[ a = 117.5 \times 0.8415 \approx 98.8 \text{ cm} \] Thus, the length of side \( a \) is approximately: \[ \boxed{98.8 \text{ cm}} \]