1. **State the problem:** We are given a triangle with angle $C = 63^\circ$, side opposite $B$ is $5.5$, side opposite $A$ is $4.7$, and we want to find the measure of angle $B$ using the Law of Sines. 2. **Recall the Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a$, $b$, and $c$ are sides opposite angles $A$, $B$, and $C$ respectively. 3. **Assign known values:** - Side opposite $B$ is $b = 5.5$ - Side opposite $A$ is $a = 4.7$ - Angle $C = 63^\circ$ 4. **Find angle $A$ first using Law of Sines:** $$\frac{a}{\sin A} = \frac{c}{\sin C}$$ But we do not know $c$ or angle $A$ yet, so instead use the ratio between sides $a$ and $b$ and their opposite angles $A$ and $B$: $$\frac{a}{\sin A} = \frac{b}{\sin B}$$ We want to find $B$, so we need to find $A$ or use angle sum later. 5. **Use Law of Sines with side $a$ and angle $C$ to find side $c$:** $$\frac{a}{\sin A} = \frac{c}{\sin C}$$ But $A$ is unknown, so instead use the fact that angles sum to $180^\circ$: $$A + B + C = 180^\circ$$ 6. **Express $A$ in terms of $B$:** $$A = 180^\circ - B - 63^\circ = 117^\circ - B$$ 7. **Apply Law of Sines between sides $a$ and $b$ and angles $A$ and $B$:** $$\frac{4.7}{\sin(117^\circ - B)} = \frac{5.5}{\sin B}$$ 8. **Cross multiply:** $$4.7 \sin B = 5.5 \sin(117^\circ - B)$$ 9. **Use sine difference identity:** $$\sin(117^\circ - B) = \sin 117^\circ \cos B - \cos 117^\circ \sin B$$ 10. **Substitute:** $$4.7 \sin B = 5.5 (\sin 117^\circ \cos B - \cos 117^\circ \sin B)$$ 11. **Distribute:** $$4.7 \sin B = 5.5 \sin 117^\circ \cos B - 5.5 \cos 117^\circ \sin B$$ 12. **Group terms with $\sin B$ on one side:** $$4.7 \sin B + 5.5 \cos 117^\circ \sin B = 5.5 \sin 117^\circ \cos B$$ 13. **Factor $\sin B$:** $$\sin B (4.7 + 5.5 \cos 117^\circ) = 5.5 \sin 117^\circ \cos B$$ 14. **Divide both sides by $\cos B$ and the factor:** $$\frac{\sin B}{\cos B} = \frac{5.5 \sin 117^\circ}{4.7 + 5.5 \cos 117^\circ}$$ 15. **Recall $\tan B = \frac{\sin B}{\cos B}$, so:** $$\tan B = \frac{5.5 \sin 117^\circ}{4.7 + 5.5 \cos 117^\circ}$$ 16. **Calculate values:** $$\sin 117^\circ \approx 0.8910, \quad \cos 117^\circ \approx -0.45399$$ 17. **Substitute:** $$\tan B = \frac{5.5 \times 0.8910}{4.7 + 5.5 \times (-0.45399)} = \frac{4.9005}{4.7 - 2.4969} = \frac{4.9005}{2.2031} \approx 2.224$$ 18. **Find angle $B$:** $$B = \arctan(2.224) \approx 65.7^\circ$$ **Final answer:** $$\boxed{B \approx 65.7^\circ}$$