Chapter 13

\(\sin \frac{B-C}{2}=\frac{b-c}{a} \cos \frac{A}{2}\)

\(b^{2} \sin 2 C+c^{2} \sin 2 B=2 b c \sin A\)

\(a(b \cos C-c \cos B)=b^{2}-c^{2}\)

\((b+c) \cos A+(c+a) \cos B+(a+b) \cos C=a+b+c\)

\(a(\cos B+\cos C)=2(b+c) \sin ^{2} \frac{A}{2}\)

\(a(\cos C-\cos B)=2(b-c) \cos ^{2} \frac{A}{2}\)

\(\frac{\sin (B-C)}{\sin (B+C)}=\frac{b^{2}-c^{2}}{a^{2}}\)

\(\frac{a+b}{a-b}=\tan \frac{A+B}{2} \cot \frac{A-B}{2}\)

\(\frac{a^{2} \sin (B-C)}{\sin B+\sin C}+\frac{b^{2} \sin (C-A)}{\sin C+\sin A}+\frac{c^{2} \sin (A-B)}{\sin A+\sin B}=0 .\)

\((b+c-a)\left(\cot \frac{B}{2}+\cot \frac{C}{2}\right)=2 a \cot \frac{A}{2}\).

\(a^{2}+b^{2}+c^{2}=2(b c \cos A+c a \cos B+a b \cos C)\)

\(\left(-a^{2}+b^{2}+c^{2}\right) \tan A=\left(a^{2}-b^{2}+c^{2}\right) \tan B=\left(a^{2}+b^{2}-c^{2}\right) \tan C .\)

\(a \sin (B-C)+b \sin (C-A)+c \sin (A-B)=0 .\)

\(\frac{a \sin (B-C)}{b^{2}-c^{2}}=\frac{b \sin (C-A)}{c^{2}-a^{2}}=\frac{c \sin (A-B)}{a^{2}-b^{2}} .\)

\(a \sin \frac{A}{2} \sin \frac{B-C}{2}+b \sin \frac{B}{2} \sin \frac{C-A}{2}+c \sin \frac{C}{2} \sin \frac{A-B}{2}=0 .\)

\(a^{2}\left(\cos ^{2} B-\cos ^{2} C\right)+b^{2}\left(\cos ^{2} C-\cos ^{2} A\right)+c^{2}\left(\cos ^{2} A-\cos ^{2} B\right)=0 .\)

\(\frac{b^{2}-c^{2}}{a^{2}} \sin 2 A+\frac{c^{2}-a^{2}}{b^{2}} \sin 2 B+\frac{a^{2}-b^{2}}{c^{2}} \sin 2 C=0 .\)

\(\frac{(a+b+c)^{2}}{a^{2}+b^{2}+c^{2}}=\frac{\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}}{\cot A+\cot B+\cot C}\)

\(a^{3} \cos (B-C)+b^{3} \cos (C-A)+c^{3} \cos (A-B)=3 a b c\)

\(a^{2} \cos 2 B+b^{2} \cos 2 A+2 a b \cos (A-B)=c^{2}\)

\((a+b+c)(\cos A+\cos B+\cos C)=2\left(a \cos ^{2} \frac{A}{2}+b \cos ^{2} \frac{B}{2}+c \cos ^{2} \frac{C}{2}\right)\)

\(\left(b^{2}-c^{2}\right) \cot A+\left(c^{2}-a^{2}\right) \cot B+\left(a^{2}-b^{2}\right) \cot C=0 .\)

\(a^{2}=(b-c)^{2}+4 b c \sin ^{2} \frac{A}{2}\)

\(\frac{1+\cos (A-B) \cos C}{1+\cos (A-C) \cos B}=\frac{a^{2}+b^{2}}{a^{2}+c^{2}}\).

\(a(\cos B \cos C+\cos A)=b(\cos C \cos A+\cos B)=c(\cos A \cos B+\cos C)\)

\(\frac{c}{a-b}=\frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{\tan \frac{A}{2}-\tan \frac{B}{2}}\)

\(\frac{c}{a+b}=\frac{1-\tan \frac{A}{2} \tan \frac{B}{2}}{1+\tan \frac{A}{2} \tan \frac{B}{2}}\)

\(\frac{a^{2} \sin B \sin C}{2 \sin A}=\Delta\).

\(\frac{s}{\cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}}=2 \sqrt[3]{\frac{a b c}{\sin A \sin B \sin C}}\)

\(\frac{1}{(b+c)^{2}} \cos ^{2}\left(\frac{B-C}{2}\right)+\frac{1}{(b-c)^{2}} \sin ^{2}\left(\frac{B-C}{2}\right)=\frac{1}{a^{2}}\)

\((b-c) \cot \frac{A}{2}+(c-a) \cot \frac{B}{2}+(a-b) \cot \frac{C}{2}=0 .\)

\(a^{2}-2 a b \cos \left(60^{\circ}+C\right)=c^{2}-2 b c \cos \left(60^{\circ}+A\right)\)

\(a^{3} \sin (B-C)+b^{3} \sin (C-A)+c^{3} \sin (A-B)=0 .\)

\(\frac{b^{2}-c^{2}}{\cos B+\cos C}+\frac{c^{2}-a^{2}}{\cos C+\cos A}+\frac{a^{2}-b^{2}}{\cos A+\cos B}=0 .\)

\(\frac{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}{4 b^{2} c^{2}}=\sin ^{2} A\)

\((a+b+c)\left(\tan \frac{A}{2}+\tan \frac{B}{2}\right)=2 c \cot \frac{C}{2}\).

\(\left(\cot \frac{A}{2}+\cot \frac{B}{2}\right)\left(a \sin ^{2} \frac{B}{2}+b \sin ^{2} \frac{A}{2}\right)=c \cot \frac{C}{2}\).

\(1-\tan \frac{A}{2} \tan \frac{B}{2}=\frac{2 c}{a+b+c} .\)

\(2 a b c \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}=2 s \Delta\)

\(b c \cos ^{2} \frac{A}{2}+c a \cos ^{2} \frac{B}{2}+a b \cos ^{2} \frac{C}{2}=s^{2}\)

\(\frac{b-c}{a} \cos ^{2} \frac{A}{2}+\frac{c-a}{b} \cos ^{2} \frac{B}{2}+\frac{a-b}{c} \cos ^{2} \frac{C}{2}=0 .\)

\(\frac{2}{(a-b)(a-c)}+\frac{2}{(b-c)(b-a)}+\frac{2}{(c-a)(c-b)}=\frac{1}{\Delta}\)

\(\sin ^{3} A \cos (B-C)+\sin ^{3} B \cos (C-A)+\sin ^{3} C \cos (A-B)=3 \sin A \sin B \sin C .\)

\(a^{3} \cos B \cos C+b^{3} \cos C \cos A+c^{3} \cos A \cos B=a b c(1-2 \cos A \cos B \cos C)\)

Solve the triangle, given i. \(\quad a=\sqrt{3}, b=\sqrt{2}\) and \(c=\frac{\sqrt{6}+\sqrt{2}}{2}\).\ ii. \(\quad b=\sqrt{3}, c=1\) and \(A=30^{\circ} .\)\ iii. \(a=5, b=7\) and \(A=60^{\circ}\). iv. \(a=1, c=2\) and \(A=30^{\circ}\). v. \(\quad a=2, c=\sqrt{3}+1\) and \(A=45^{\circ}\).= vi. \(a=\sqrt{3}, b=\sqrt{2}\) and \(A=60^{\circ} .\) vii. \(a=4, b=5\) and \(A=120^{\circ}\). ix. \(\quad a=2, B=60^{\circ}\) and \(C=45^{\circ} .\) x. \(A=45^{\circ}, B=60^{\circ}\) and \(C=75^{\circ} .\)

In a \(\triangle A B C\), if \(A=45^{\circ}, b=\sqrt{6}, a=2\), then find \(B\).

In triangle \(A B C, A=30^{\circ}, b=8, a=6\), then find \(B\).

If \(A=30^{\circ}, a=7, b=8\) in \(\Delta A B C\), then how many values of \(B\) are possible?

If the data given to construct a triangle \(A B C\) are \(a=5, b=7, \sin A=\frac{3}{4}\), then how many triangles can be constructed?

In a triangle whose sides are 3,4 and \(\sqrt{38}\) meters respectively, prove that the largest angle is greater then \(120^{\circ} .\)