B3. Three circles with the same centre $O$ and areas $2 \pi, 3 \pi$ and $4 \pi$ are drawn. From a point $A$ on the largest circle, tangent lines are drawn to points $B$ on the middle circle and $C$ on the smallest circle. If $B, C$ are on the same side of $O A$, find the exact value of $\angle B A C$.
Solution: Draw segments $O B$ and $O C$. Note that $A B O$ and $A C O$ are right angles. The line segment $O B$ is the radius of a circle with area $3 \pi$, so it has length $O B=\sqrt{3}$. Similarly, the line segment $O C$ is the radius of a circle with area $2 \pi$, so it has length $O C=\sqrt{2}$, and $O A$ has length 2 .
From $A B O, \sin \angle B A O=\frac{\sqrt{3}}{2}$, so $\angle B A O=60^{\circ}$.
From $A C O, \sin \angle C A O=\frac{\sqrt{2}}{2}$ so $\angle C A O=45^{\circ}$.
Then $\angle B A C=\angle B A O-\angle C A O=60^{\circ}-45^{\circ}=15^{\circ}$.
Answer: $15^{\circ}$