For the triangle ABC , with usual notations, if the angles \(A, B, C\) are in A.P. and \(\mathrm{m} \angle \mathrm{A}=30^{\circ}, \mathrm{c}=3\), then the values of a and b are respectively

Angles A, B, C are in A.P.
\(\therefore \angle \mathrm{A}+\angle \mathrm{C}=2 \angle \mathrm{~B} \)
\( \mathrm{Also}, \angle \mathrm{~A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ} \)
\( 2 \angle \mathrm{~B}+\angle \mathrm{B}=180^{\circ} \)
\( \therefore \angle \mathrm{B}=60^{\circ} \)
\( \angle \mathrm{A}=30^{\circ} ...[Given]\)
\( \therefore \angle \mathrm{C}=90^{\circ}\)
Using sine Rule
\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \)
\( \therefore \frac{a}{\sin 30}=\frac{b}{\sin 60}=\frac{3}{\sin 90} \)
\( \therefore \frac{a}{\frac{1}{2}}=\frac{b}{\frac{\sqrt{3}}{2}}=\frac{3}{1} \)
\( \Rightarrow 2 a=3, \frac{2 b}{\sqrt{3}}=3\)
\(\Rightarrow \mathrm{a}=\frac{3}{2} \)
\( \Rightarrow \mathrm{~b}=\frac{3 \sqrt{3}}{2}\)