If the angles $A, B$ and $C$ of a triangle are in an arithmetic progression and if $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively, then the value of the expression $\frac{a}{c} \sin 2 C+\frac{c}{a} \sin 2 A$ is

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Aicharya · Accepted Answer

Since, $A, B, C$ are in $\mathrm{AP}$ $ \begin{aligned} & \Rightarrow \quad 2 B=A+C 	ext { ie, } \angle B=60^{\circ} \ & 	herefore \frac{a}{c}(2 \sin C \cos C)+\frac{c}{a}(2 \sin A \cos A) \ & =2 k(a \cos C+c \cos A) \ & \quad\left[\begin{array}{l} 	ext { using, } \ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=\frac{1}{k} \end{array}ight] \ & =2 k(b) \ & =2 \sin B \end{aligned} $ $ =\sqrt{3} $ $ 	ext { [using, } b=a \cos C+c \cos A] $

If the angles $A, B$ and $C$ of a triangle are in an arithmetic progression and if $a, b$ and $c$ denote the…

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