If $A$ is the area and $2s$ is the sum of $3$ sides of a triangle,then:

$A$ tower $AB$ leans towards west making an angle $\alpha$ with the vertical. The angular elevation of $B$,the top most point of the tower,is $\beta$ as observed from a point $C$ due east of $A$ at a distance $d$ from $A$. If the angular elevation of $B$ from a point $D$ due east of $C$ at a distance $2d$ from $C$ is $\gamma$,then $2\tan \alpha$ can be given as

In $\triangle PQR$,let $\angle P > \angle Q$. If the radian measures of $\angle P$ and $\angle Q$ satisfy the equation $4 \sin^3 x - 3 \sin x + a = 0$ where $0 < a < 1$,then the radian measure of $\angle R$ is

With usual notation in a $\Delta ABC$,$\left( \frac{1}{r_1} + \frac{1}{r_2} \right) \left( \frac{1}{r_2} + \frac{1}{r_3} \right) \left( \frac{1}{r_3} + \frac{1}{r_1} \right) = \frac{K R^3}{a^2 b^2 c^2}$ where $K$ has the value equal to

In a triangle with one of the angles $120^{\circ}$,the lengths of the sides form an $A$.$P$. If the length of the greatest side is $7 \ m$,then the area of the triangle is

With usual notations in a triangle $ABC$,the product $(I I_1) \cdot (I I_2) \cdot (I I_3)$ has the value equal to