In a triangle $ABC$,$\sin A : \sin B : \sin C = 1 : 2 : 3$. If $b = 4 \, \text{cm}$,the perimeter of the triangle is ..... $\text{cm}$.

In triangle $ABC$,if $a=13, b=8, c=7$,then $\cos(B+C) = $

In $\triangle ABC$,if $a=2(\sqrt{3}+1)$,$B=45^{\circ}$ and $C=60^{\circ}$,then the area (in sq.units) of that triangle is

If in a triangle $\sin A : \sin C = \sin (A - B) : \sin (B - C)$,then $a^2, b^2, c^2$:

If in a $\triangle ABC$,$\frac{1}{a+c} + \frac{1}{b+c} = \frac{3}{a+b+c}$,then $\angle C$ is equal to (in $^{\circ}$)

The sides of a triangle are $\sin \alpha$,$\cos \alpha$,and $\sqrt{1 + \sin \alpha \cos \alpha}$ for some $0 < \alpha < \frac{\pi}{2}$. Then the greatest angle of the triangle is.....$^o$