1. The problem is to find angle $C$ in a triangle using the sine rule, then calculate the remaining angle using $180 - 40 + C$. 2. The sine rule states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a,b,c$ are sides opposite angles $A,B,C$ respectively. 3. Given angle $A = 40^\circ$ and side $a$, and side $b$, we use the sine rule to find angle $C$: $$\sin C = \frac{c \sin A}{a}$$ 4. Calculate $C$ by taking the inverse sine: $$C = \sin^{-1}\left(\frac{c \sin 40^\circ}{a}\right)$$ 5. Finally, find the remaining angle using: $$180^\circ - 40^\circ - C$$ This gives the third angle of the triangle. Note: The exact values of sides $a$ and $c$ are needed to compute $C$ numerically.