The sides of a $\triangle ABC$ are given by $a=3, b=5$ and $c=3$. Then,$\cos A=$

In a $\triangle ABC$,if $\angle A = 60^{\circ}$,then $\frac{b}{c+a} + \frac{c}{a+b} = $

If in a $\Delta ABC$,the altitudes from the vertices $A, B, C$ on opposite sides are in $H.P.$,then $\sin A, \sin B, \sin C$ are in

The perimeter of a $\triangle ABC$ is $6$ times the arithmetic mean of the values of the sine of its angles. If its side $BC$ is of unit length,then $\angle A=$

If $p_1, p_2, p_3$ are altitudes of a triangle $ABC$ from the vertices $A, B, C$ respectively and if $\Delta$ is the area of the triangle,$S$ is the semi-perimeter of the triangle,then find the value of $\frac{\cos A}{p_1} + \frac{\cos B}{p_2} + \frac{\cos C}{p_3}$.

The base of a triangle is $80$ and one of the base angles is $60^{\circ}$. If the sum of the lengths of the other two sides is $90$,then the shortest side is of length