If the sides of a triangle $ABC$ whose perimeter is $42$ are in arithmetic progression,its circum-radius is $\frac{65}{8}$ and $B < A < C$,then $\sin A=$

If $2 \cos \theta + \sin \theta = 1$,then the value of $4 \cos \theta + 3 \sin \theta$ is equal to

Consider an obtuse-angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius $1$.
$(1)$ Let $a$ be the area of the triangle $ABC$. Then the value of $(64 a)^2$ is
$(2)$ The inradius of the triangle $ABC$ is

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 $\frac{a}{c} \sin 2C + \frac{c}{a} \sin 2A$ is

In $\triangle ABC$,$(a-b)^2 \sin^2\left(\frac{A+B}{2}\right) + (a+b)^2 \sin^2\left(\frac{C}{2}\right) = $

With usual notations,in $\triangle ABC$,if $a=2, b=3, c=5$ and $\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{k+7}{30}$,then $k=$