Animated Solution for Mathematics - Trigonometry: Let the angles $A,B,C$ of a triangle $ABC$ be in A.P. and let $b:c=\sqrt{3}:\sqrt{2}$. Find the angle $A$.

In a triangle $ABC$, the angles $A,B,$ and $C$ are in an Arithmetic Progression (A.P.). By the definition of an A.P., the middle term is the average of its neighbors, which gives us the relation:

We also know the fundamental property of Euclidean geometry that the sum of the interior angles of any triangle is $180^{\circ }$:

By substituting the A.P. condition ($A+C=2B$) into the angle sum equation, we obtain:

This confirms that in any triangle where the angles are in A.P., the middle angle must always be $60^{\circ }$.

We are given the ratio of sides $b:c=\sqrt{3}:\sqrt{2}$. To relate these sides to the angles, we utilize the Sine Rule:

Rearranging this to solve for $\sin C$, we get:

Substituting the known values $B=60^{\circ }$, $b=\sqrt{3}$, and $c=\sqrt{2}$:

The equation $\sin C=\frac{1}{\sqrt{2}}$ implies that $C$ could be $45^{\circ }$ or $135^{\circ }$.

We must check the validity of these values against the triangle constraint $A+B+C=180^{\circ }$. If $C=135^{\circ }$, then $B+C=60^{\circ }+135^{\circ }=195^{\circ }$, which exceeds the total sum of angles in a triangle.

Therefore, we must have $C=45^{\circ }$. Finally, we calculate angle $A$:

The angles of the triangle are $75^{\circ },60^{\circ },$ and $45^{\circ }$.