1. **State the problem:** We have triangle $\triangle ABC$ with sides $AB = 3.8$ cm, $BC = 5.2$ cm, and angle $\angle ABC = 35^\circ$. We want to find $\sin(C)$. 2. **Identify given information:** Here, side $a = BC = 5.2$ cm, side $c = AB = 3.8$ cm, and angle $B = 35^\circ$. 3. **Apply the Law of Sines:** The Law of Sines states $$\frac{a}{\sin A} = \frac{c}{\sin C}$$ We know $a$, $c$, and $\sin A$ (since $A = 35^\circ$ and $\sin 35^\circ \approx 0.57$). 4. **Rearrange to find $\sin C$:** $$\sin C = \frac{c \cdot \sin A}{a}$$ 5. **Substitute values:** $$\sin C = \frac{3.8 \times 0.57}{5.2}$$ 6. **Calculate:** $$\sin C = \frac{2.166}{5.2} \approx 0.4165$$ 7. **Conclusion:** The value of $\sin(C)$ is approximately $0.42$.