1. **State the problem:** We are given that angles $A$ and $B$ are complementary, meaning their measures add up to 90 degrees, and that $m \angle B = 55^\circ$. We want to verify if $\sin(B) = \sin(A)$. 2. **Recall the definition of complementary angles:** Two angles are complementary if $m \angle A + m \angle B = 90^\circ$. 3. **Find $m \angle A$:** Since $m \angle B = 55^\circ$, then $$m \angle A = 90^\circ - m \angle B = 90^\circ - 55^\circ = 35^\circ.$$ 4. **Use the sine function:** We want to check if $\sin(55^\circ) = \sin(35^\circ)$. 5. **Recall the sine complementary angle identity:** For any angle $\theta$, $$\sin(90^\circ - \theta) = \cos(\theta).$$ 6. **Apply the identity:** Since $A = 90^\circ - B$, $$\sin(A) = \sin(90^\circ - B) = \cos(B).$$ 7. **Compare $\sin(B)$ and $\sin(A)$:** We have $$\sin(B) \neq \cos(B)$$ in general, so $$\sin(B) \neq \sin(A).$$ **Final answer:** $\sin(B) \neq \sin(A)$ when $A$ and $B$ are complementary and $m \angle B = 55^\circ$.