In a $\triangle A B C$, if $a=5, b=3, c=7$, then $\sqrt{\frac{\sin (A-B)}{\sin (A+B)}}=$

In a $\triangle A B C$, if $r_{1}=6, r_{2}=9, r_{3}=18$, then $\cos A=$

If $A B C$ is an isosceles triangle with base $B C$, then $r_{1}=$

In $\triangle A B C$, if $r_{1}+r_{2}=3 R, r_{2}+r_{3}=2 R$, then

In a $\triangle A B C$, the sides $b, c$ are fixed. In measuring angle $A$, if there is an error of $\delta A$, then the percentage error in measuring the length of the side $a$ is

In triangle $A B C$, if $a=4, b=3$ and $c=2$, then $2(a-b \cos C)(a-c \sec B)=$

In $\triangle A B C$, if $A=45^{\circ}, C=75^{\circ}$ and $R=\sqrt{2}$, than $r=$

If $A(1,2,-3), B(2,3,-1)$ and $C(3,1,1)$ are the vertices of $\triangle A B C$, then $\left|\frac{-\cos A}{\cos B}\right|=$

If $A+B+C=2 S$, then $\sin (S-A) \cos (S-B)-\sin (S-C) \cos S=$

In a $\triangle A B C$, if $\tan \frac{A}{2}: \tan \frac{B}{2}: \tan \frac{C}{2}=15: 10: 6$, then $\frac{a}{b-c}=$

In a $\triangle A B C, \frac{a\left(r_1+r_2 r_3\right)}{r_1-r+r_2+r_3}=$

In a $\triangle A B C$, if $(a-b)^2 \cos ^2 \frac{C}{2}+(a+b)^2 \sin ^2 \frac{C}{2}=a^2+b^2$, then $\cos A=$

In a $\triangle A B C$, if $r_1 r_2+r_3=35, r_2 r_3+r_1=63$ and $r_3 r_1+r_2=45$, then $2 s=$

In an isosceles right angled triangle, a straight line is drawn from the mid-point of one of the equal sides to the opposite vertex. Then, a pair of possible values of the cotangents of the two angles so formed at that vertex are

In a $\triangle A B C$, if $r_1=2 r_2=3 r_3$, then $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=$