Question

If angles of a triangle are in A.P. and $b:c=\sqrt{3}:\sqrt{2},$ then C = ______________.

Answer 1

If angle of a triangle ABC are in A.P
⇒ 2∠B = ∠A + ∠C

and $\frac{b}{c}=\frac{\sqrt{3}}{\sqrt{2}}$
By angle sum property
∠A + ∠B + ∠C = $\mathrm{\pi }$
⇒ 2∠B + ∠B = $\mathrm{\pi }$
⇒ ∠B = $\frac{\mathrm{\pi }}{3}$
also,
Using Sine formula
$$\begin{gathered} \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \\ \mathrm{i}.\mathrm{e}\frac{b}{\sin B}=\frac{c}{\sin C} \\ \mathrm{i}.\mathrm{e}\frac{b}{c}=\frac{\sin B}{\sin C} \\ \mathrm{i}.\mathrm{e}\frac{\sqrt{3}}{\sqrt{2}}=\frac{\sin {\displaystyle \frac{\mathrm{\pi }}{3}}}{\sin C}\left(\because \frac{b}{c}=\frac{\sqrt{3}}{\sqrt{2}}\mathrm{given}\right) \\ \mathrm{i}.\mathrm{e}\sin C=\frac{\sqrt{3}}{2}\times \frac{\sqrt{2}}{\sqrt{3}} \\ \mathrm{i}.\mathrm{e}\sin C=\frac{1}{\sqrt{2}} \\ \mathrm{i}.\mathrm{e}\sin C=\frac{\mathrm{\pi }}{4} \end{gathered}$$