find side length using law of sines

1. Identify the given information: From the triangle, we have angle $A = 46.7^\circ$ and its opposite side $a = 44$. We also have angle $C = 48^\circ$ and its opposite side $b$, which we need to find.

2. State the Law of Sines: The Law of Sines states that for any triangle with sides $a, b, c$ and opposite angles $A, B, C$:
$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$
In this case, we use the relationship between side $a$, angle $A$, side $b$, and angle $C$:
$$\frac{a}{\sin(A)} = \frac{b}{\sin(C)}$$

3. Substitute the known values: Plug the values into the formula:
$$\frac{44}{\sin(46.7^\circ)} = \frac{b}{\sin(48^\circ)}$$

4. Solve for b: Multiply both sides by $\sin(48^\circ)$:
$$b = \frac{44 \cdot \sin(48^\circ)}{\sin(46.7^\circ)}$$

5. Calculate the result: Using a calculator:
$$\sin(48^\circ) \approx 0.74314$$
$$\sin(46.7^\circ) \approx 0.72777$$
$$b \approx \frac{44 \cdot 0.74314}{0.72777} \approx \frac{32.698}{0.72777} \approx 44.929$$

1. Round to the nearest tenth: Rounding $44.929$ to the nearest tenth gives $44.9$.