KC Sinha Mathematics Solution Class 11 Chapter 9 Sine and Cosine Formulae and Their Applications Exercise 9.1

 Exercise 9.1

 Page no 9.32

Type -1 

Question 1

If the angles of a $\triangle A B C$ are in the ratio $2: 3: 7$, show that $a: b: c=\sqrt{2}: 2:(\sqrt{3}+1)$

Question 2

If in $\triangle A B C a=4, b=12, \angle B=30^{\circ}$ then find $\sin A$

Question 3

In any $\Delta A B C$, prove that $\frac{\sin B}{\sin (B+C)}=\frac{b}{a}$

Question 4

If the two angles of a triangle are $30^{\circ}$ and $45^{\circ}$ and the included side is $(\sqrt{3}+1)$ $\mathrm{cm}$, find the other sides of the triangle.

Question 5

The angles of a triangle are as $7: 2: 1$; prove that the ratio of the least side to the greatest is $3-\sqrt{5}: 2$

Question 6

In $\triangle A B C$, if $c=3.4 \mathrm{~cm}, A=25^{\circ}, B=85^{\circ}$, find $a, b$ and angle $C$. $\left(\sin 25^{\circ}=0.4226, \sin 85^{\circ}=0.9962\right)$

Type -2

Question 7

Find the greatest angle of $\triangle A B C$ if

(i) $a=2, b=\sqrt{6}, c=\sqrt{3}-1$

(ii) $a=m, b=n, c=\sqrt{m^{2}+m n+n^{2}}$

Question 8

In $\triangle A B C$, if $a=25, b=52$ and $c=63$, find $\cos A$.

Question 9

In a $\triangle A B C$, if $a=18, b=24, c=30$, find $\cos A, \cos B, \cos C$.

Question 10

In any $\triangle A B C$ if $B=60^{\circ}$, then prove that $(c-a)^{2}=b^{2}-c a$.

Question 11

If the sides of a triangle are 3,5 and 7, prove that the triangle is obtuse angled triangle and find the obtuse angle.

Question 12

In any $\triangle A B C$, prove that

$\frac{b^{2}+c^{2}-a^{2}}{c^{2}+a^{2}-b^{2}}=\frac{\tan B}{\tan A}$

Question 13

If in a $\triangle A B C, c^{4}-2\left(a^{2}+b^{2}\right) c^{2}+a^{4}+a^{2} h^{2}+h^{4}=a$ prove that $C=60^{2}$ or $120^{\circ}$.

Question 14

In a $\triangle A B C$. if $a^{4}+b^{4}+c^{4}=2 c^{2}\left(a^{2}+b^{2}\right)$, prove that $C=45^{\circ}$ or $135^{\circ}$

Page no 9.33

Type -3

Question 15

In $\triangle A B C$, prove that

(i) $a(\cos B+\cos C-1)+b(\cos C+\cos A-1)+c(\cos A+\cos B-1)=0$

(ii) $a \cos (A+B+C)-b \cos (B+A)-c \cos (A+C)=0$

Question 16

In any $\triangle A B C$, prove that

$\frac{c-b \cos A}{b-c \cos A}=\frac{\cos B}{\cos C}$

Question 17

In $\triangle A B C, a=4, b=6, c=8$, then find the value of $8 \cos A+16 \cos B+4 \cos C$

Type -4

Question 18

In any $\triangle A B C$, prove that

(i) $(b+c) \cos \frac{B+C}{2}=a \cos \frac{B-C}{2}$

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

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

(iv) If $b+c=2 a \cos \frac{B-C}{2}$, then prove that $A=60^{\circ}$

Question 19

In any $\triangle A B C$, prove that

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

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

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

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

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

(vi) $\tan \left(\frac{A}{2}+B\right)=\frac{c+b}{c-b} \tan \frac{A}{2}$

Question 20

In any $\triangle A B C$, prove that

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

Question 21

If $A=2 B$, then prove that either $c=b$ or $a^{2}=b(c+b)$ [Hint : $A=2 B \Rightarrow \sin A=\sin 2 B$ ]

Page no 9.34

Question 22

In any $\triangle A B C$, prove that

$\frac{\cos A}{a}+\frac{a}{b c}=\frac{\cos B}{b}+\frac{b}{c a}=\frac{\cos C}{c}+\frac{c}{a b}$

Question 23

In $\triangle A B C$ if $A+C=2 B$, prove that $2 \cos \frac{A-C}{2}=\frac{a+c}{\sqrt{a^{2}-a c+c^{2}}}$

Question 24

In any $\triangle A B C$, prove that

(i) $\frac{\cos 2 A}{a^{2}}-\frac{\cos 2 B}{b^{2}}=\frac{1}{a^{2}}-\frac{1}{b^{2}}$

(ii) $\left(b^{2}-c^{2}\right) \sin ^{2} A+\left(c^{2}-a^{2}\right) \sin ^{2} B+\left(a^{2}-b^{2}\right) \sin ^{2} C=0$

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

Question 25

If the median $A D$ of $\triangle A B C$ be perpendicular to side $A B$ then prove that $\tan A+2 \tan B=0$

[Hint : $A D=B D \sin B$ and $\frac{C D}{\sin \left(A-90^{\circ}\right)}=\frac{A D}{\sin C}$

$\therefore \sin C+\cos A \sin B=0]$

Question 26

In a $\triangle A B C$, if $\tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2}$ are in $A \cdot P$, prove that $\cos A, \cos B, \cos C$ are also in $A . P$

Question 27

If $\frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)}$, prove that $a^{2}, b^{2}, c^{2}$ are in $A \cdot P$

Type -5

Question 28

If in $\triangle A B C, a=15, b=36, c=39$ find $\tan \frac{A}{2}$

Question 29

In a $\triangle A B C$, if $a=18, b=24, c=30$, find

(i) $\tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2}$

(ii) $\sin A, \sin B, \sin C$

Question 30

In any $\triangle A B C$, prove that

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

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

Question 31

In $\triangle A B C$, prove that

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

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

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

Page no 9.35

Question 32

In any $\triangle A B C$, prove that : $\frac{\cos ^{2} \frac{A}{2}}{a}+\frac{\cos ^{2} \frac{B}{2}}{b}+\frac{\cos ^{2} \frac{C}{2}}{c}=\frac{s^{2}}{a b c}$

Question 33

In $\triangle A B C$, prove that

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

(ii) $\tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}=\sqrt{\left(1-\frac{a}{s}\right)\left(1-\frac{b}{s}\right)\left(1-\frac{c}{s}\right)}$

Question 34

If in a $\Delta A B C, \sin A, \sin B, \sin C$ are in A. P., show that $3 \tan \frac{A}{2} \tan \frac{C}{2}=1$ [Hint : $\sin A, \sin B, \sin C$ are in A.P. $\Rightarrow a, b, c$ are in A.P.]

Question 35

In any $\triangle A B C$, prove that $(b+c-a)\left(\cot \frac{B}{2}+\cot \frac{C}{2}\right)=2 a \cot \frac{A}{2}$

Type -6

Question 36

In a $\triangle A B C$, the sum of two sides is $\sqrt{3}$ times their difference and the included angle is $60^{\circ}$; find the difference of the remaining angles.

Question 37

If in $\triangle A B C$, the difference of two angles is $60^{\circ}$ and the remaining angle is $30^{\circ}$, then find the ratio of the sides opposite to first two angles.

Type -7

Question 38

In a $\triangle A B C$, if $a=18, b=24, c=30$, find the area of $\triangle A B C$.

Question 39

The sides of a triangle are in A. P. Its area is $\frac{3}{5}$ th of an equilateral triangle of

the same perimeter, show that the sides are in the proportion $3: 5: 7$.

Question 40

The sides of a quadrilateral are $3,4,5$ and $6 \mathrm{cms}$. The sum of a pair of opposite angles is $120^{\circ}$. Show that the area of the quadrilateral is $3 \sqrt{30} \mathrm{sq} . \mathrm{cm}$.