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

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 $a = 9, b = 8$ and $c = x$ satisfies $3 \cos C = 2$,then

In a triangle,the sum of lengths of two sides is $x$ and the product of the lengths of the same two sides is $y$. If $x^2 - c^2 = y$,where $c$ is the length of the third side of the triangle,then the circumradius of the triangle is

If $S$ is the circumcentre of a triangle $ABC$, $a=5, b=6, c=9$ and $SB=\frac{27}{4 \sqrt{2}}$, then $\sin 2C=$

If the angles of a triangle are in the ratio $1 : 2 : 7,$ then the ratio of its greatest side to the least side is