If the lengths of the sides of a triangle are $3, 5, 7,$ then the largest angle of the triangle is

In a $\Delta ABC$,the expression $\left( \frac{a^2}{\sin A} + \frac{b^2}{\sin B} + \frac{c^2}{\sin C} \right) \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$ simplifies to,where $\Delta$ is the area of the triangle.

In any $\triangle ABC$,the expression $\frac{b-c \cos A}{c-b \cos A}$ is equal to:

In $\triangle ABC$,if $a = 2b$ and $|A - B| = \frac{\pi}{3}$,then $\angle C = $

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

If $R$ is the circumradius of $\triangle ABC$,then the area of $\triangle ABC$ is equal to $\ldots$.