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KC Sinha Mathematics Solution Class 11 Chapter 8 Sine and Cosine के अनुप्रयोग (Application of Sine and Cosine) Exercise 8.1

Exercise 8.1

Question 1

Sol :
∠A+∠B+∠C=180°

2x+3x+7x=180°

12x=180°

x=\frac{180^{\circ}}{12}

∠A=2x=2×15=30°

∠B=3x=3×15=45°

∠C=7x=7×15=105°

sine formula:

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB , c=ksinC

a=ksin30° , b=ksin45° , c=ksin105°

a=\frac{k}{2}b=\frac{k}{\sqrt{2}} , c=ksin(90°+15°)

c=kcos15°

c=kcos(45°-30°)

c=k(cos45°cos30°+sin45°sin30°)

c=k\left(\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}} \times \frac{1}{2}\right)

c=k \frac{(\sqrt{3}+1)}{2 \sqrt{2}}


\frac{a}{b}=\frac{\frac{k}{2}}{\frac{k}{\sqrt{2}}}

=\frac{\sqrt{2}}{2}


\frac{b}{c}=\frac{\frac{k}{\sqrt{2}}}{k \frac{(\sqrt{3}+1)}{2 \sqrt{2}}}

=\frac{2}{\sqrt{3}+1}


∴a:b:c=√2:2:(√3+1)


Question 2

Sol :
\frac{a}{\sin A}=\frac{b}{\sin B}

\frac{4}{\sin A}=\frac{12}{\sin 30^{\circ}}

\frac{4}{\sin A}=\frac{12}{\frac{1}{2}}

\frac{4}{\sin 2 x}=24

\frac{4}{24}=\sin A

\frac{1}{6}=\sin A


Question 3

Sol :
Sine formula:
\frac{a}{\sin A}=\frac{b}{\sin B}

\frac{a}{\sin [180^{\circ}-(B+C)]}=\frac{b}{\sin B}

\frac{a}{\sin (B+C)}=\frac{b}{\sin B}

\frac{\sin B}{\sin (B+C)}=\frac{b}{a}


Question 4

Sol :
∠A+∠B+∠C=180°

30°+45°+∠C=180°

75°+∠C=180°

∠C=105°

sine formula:
\frac{a}{\sin a}=\frac{b}{\sin b}=\frac{c}{\sin c}

\frac{a}{\sin 30^{\circ}}=\frac{b}{\sin 45^{\circ}}=\frac{\sqrt{3}+1}{\sin 105^{\circ}}

\frac{a}{\frac{1}{2}}=\frac{b}{\frac{1}{\sqrt{2}}}=\frac{\sqrt{3}+1}{\frac{\sqrt{3}+1}{2 \sqrt{2}}}

2a=b√2=2√2


CASE-1
2a=2√2

a=√2

CASE-II
b√2=2√2

b=2


Question 5

Sol :
∠A+∠B=60°..(i) ∠C=30°

∠A+∠B+∠C=180°

∠A+∠B+30°=180°

∠A+∠B=180°-30°

∠A+∠B=150°..(ii)

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∠A-∠B=60°

\begin{aligned} &\angle A-\angle B=60^{\circ}\\ &\begin{array}{r} \angle A+\angle B=150^{\circ} \\ \hline 2 \angle A=210 \end{array} \end{aligned}

∠A=105° , ∠B=45°

sine formula:
\frac{a}{\sin A}=\frac{b}{\sin B}

\frac{a}{b}=\frac{\sin A}{\sin B}

\frac{a}{b}=\frac{\sin 105^{\circ}}{\sin 45^{\circ}}

\frac{a}{b}=\frac{\frac{\sqrt{3}+1}{2 \sqrt{2}}}{\frac{1}{\sqrt{2}}}

a:b=(√3+1):2

Question 6

Sol :
∠A=7x , ∠B=2x , ∠C=x

∠A+∠B+∠C=180°

7x+2x+x=180°

10x=180°

x=18°

∠A=7x=7×18°=126°
 ∠C=x=18°

sine formula:
\frac{a}{\sin A}=\frac{c}{\sin C}

\frac{\sin c}{\sin A}=\frac{c}{a}

\frac{\sin 18^{\circ}}{\sin 126^{\circ}}=\frac{c}{a}

\frac{\sin 18^{\circ}}{\sin (90^{\circ}+36^{\circ})}=\frac{c}{a}

\frac{\sin 18^{\circ}}{\cos 36^{\circ}}=\frac{c}{a}

[∵\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}

 \cos 36^{\circ}=\frac{\sqrt{5}+1}{4}]

\frac{\frac{\sqrt{5}-1}{4}}{\frac{\sqrt{5+1}}{4}}=\frac{c}{a}

\frac{\sqrt{5}-1}{\sqrt{5}+1} \times \frac{\sqrt{5}-1}{\sqrt{5}-1}=\frac{c}{a}

\frac{5-\sqrt{5}-\sqrt{5}+1}{(\sqrt{5})^{2}-1^{2}}=\frac{c}{a}

\frac{6-2 \sqrt{3}}{5-1}=\frac{c}{a}

\frac{2(3-\sqrt{5})}{4}=\frac{c}{a}

(3-√5):2=c:a


Question 7

Sol :
\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}

\cos B=\frac{a^{2}+c^{2}-b^{2}}{2 a c}

\cos B=\frac{2^{2}+(\sqrt{3}-1)^{2}-(\sqrt{6})^{2}}{2 \times 2 \times(\sqrt{3}-1)}

\cos B=\frac{4+(\sqrt{3})^{2}+1^{2}-2 \sqrt{3} \cdot 1-6}{4(\sqrt3-1)}

\cos B=\frac{4+3+1-2 \sqrt{3}-6}{4(\sqrt{3}-1)}

\cos B=\frac{-2 \sqrt{3}+2}{4(\sqrt{3}-1)}

=\frac{-2(\sqrt{3}-1)}{4(\sqrt{3}-1)}=\frac{1}{2}

cosB=-cos60°

cosB=cos(180°-60°)

cosB=cos120°

B=120°


Question 8

Sol :
cosine formula:
\cos B=\frac{c^{2}+a^{2}-b^{2}}{2 c a}

\cos 60^{\circ}=\frac{c^{2}+a^{2}-b^{2}}{2 c a}

\frac{1}{2}=\frac{c^{2}+a^{2}-b^{2}}{2c a}

c2+a2-b2=ca

c2+a2-ca=b2

c2+a2-2ca+ca=b2

(c-a)2=b2-ca


Question 9

Sol :
L.H.S

\frac{b^{2}+c^{2}-a^{2}}{c^{2}+a^{2}-b^{2}}

[]

=\frac{\frac{b^{2}+c^{2}-a^{2}}{2 a b c}}{\frac{c^{2}+a^{2}-b^{2}}{2 a b c}}

=\frac{\frac{b^{2}+c^{2}-a^{2}}{2 b c \times a}}{\frac{c^{2}+a^{2}-b^{2}}{2 \operatorname{ca}\times b}}

=\frac{\frac{\cos A}{a}}{\frac{\cos B}{b}}

=\frac{\cos A}{a} \times \frac{b}{\cos B}

\because \left[\frac{a}{\sin A}=\frac{b}{\sin B}\right]

=\frac{\cos A}{\frac{b \sin A}{\sin B}} \times \frac{b}{\cos B}

=\frac{\cos A}{\sin A} \times \frac{\sin B}{\cos B}

=cotA×tanB

=\frac{\tan B}{\tan A}


Question 10

Sol :
a=3 , b=5 , c=7

\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}

\cos C=\frac{3^{2}+5^{2}-7^{2}}{2 \times 3 \times 5}

\cos C=\frac{9+25-49}{30}

\cos C=\frac{-15}{30}

cos C=-cos60°

cosC=cos(180°-60°)

cosC=cos(120°)

C=120°

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Question 11

Sol :

a=8 , b=10 , c=12

\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}

=\frac{10^{2}+12^{2}-8^{2}}{2 \times 10 \times 12}

=\frac{100+144-64}{240}

=\frac{180^{\circ}}{240^{\circ}}=\frac{3}{4}


\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}

=\frac{8^{2}+10^{2}-12^{2}}{2 \times 8 \times 10}

=\frac{64+100-144}{160}=\frac{20^{\circ}}{160^{\circ}}

2 \cos ^{2} \frac{c}{2}-1=\frac{1}{8}

2 \cos ^{2} \frac{c}{2}=\frac{1}{8}+1

2 \cos ^{2} \frac{c}{2}=\frac{1+8}{8}

\cos ^{2} \frac{c}{2}=\frac{9}{16}

\cos \frac{c}{2}=\pm \sqrt{\frac{9}{16}}

[\because \cos \frac{c}{2}>0]

\cos \frac{c}{2}=\frac{3}{4}

\therefore \cos A=\cos \frac{c}{2}

A=\dfrac{c}{2}

⇒2A=C


Question 12

Sol :
c^{4}-2\left(a^{2}+b^{2}\right) c^{2}+a^{4}+a^{2} b^{2}+b^{4}=0

c^{4}-2\left(a^{2}+b^{2}\right) c^{2}+\left(a^{2}\right)^{2}+2 a^{2} \cdot b^{2}+\left(b^{2}\right)^{2}-a^{2} b^{2}=0

\left(c^{2}\right)^{2}-2\left(a^{2}+b^{2}\right) c^{2}+\left(a^{2}+b^{2}\right)^{2}=a^{2} b^{2}

\left(a^{2}+b^{2}-c^{2}\right)^{2}=a^{2} b^{2}

\left(a^{2}+b^{2}-c^{2}\right)^{2}=\frac{1}{4} \times 4 a^{2} b^{2}

\frac{\left(a^{2}+b^{2}-c^{2}\right)^{2}}{4 a^{2} b^{2}}=\frac{1}{4}

\left(\frac{a^{2}+b^{2}-c^{2}}{2 a b}\right)^{2}=\frac{1}{4}

\frac{a^{2}+b^{2}-c^{2}}{2 a b}=\pm \sqrt{\frac{1}{4}}

\cos C=\pm \frac{1}{2}

CASE-I

\cos C=\frac{1}{2}

cosC=cos60°

C=60°


CASE-II

\cos C=-\frac{1}{2}

cos C=-cos60°

cos C=cos(180°-60°)

cos C=cos(120°)

C=120°


Question 13

Sol :

(a+b+c)(b+c-a)=3bc

(b+c+a)(b+c-a)=3bc

(b+c)^{2}-a^{2}=3 b c

b^{2}+c^{2}+2 b c-a^{2}=3 b c

b^{2}+c^{2}-a^{2}=b c

b^{2}+c^{2}-a^{2}=\frac{1}{2} \times 2 b c

\frac{b^{2}+c^{2}-a^{2}}{2 b c}=\frac{1}{2}

cos A=cos60°

A=60°


Question 14

Sol :
a^{4}+b^{4}+c^{4}=2 c^{2}\left(a^{2}+b^{2}\right)

a^{4}+b^{4}+c^{4}=2 c^{2} a^{2}+2 b^{2} c^{2}

a^{4}+b^{4}+c^{4}-2 b^{2} c^{2}-2 c^{2} a^{2}=0

[]

a^{4}+b^{4}+c^{4}+2 a^{2} b^{2}-2 b^{2} c^{2}-2 c^{2} a^{2}=2 a^{2} b^{2}

\left(a^{2}\right)^{2}+\left(b^{2}\right)^{2}+\left(c^{2}\right)^{2}+2 a^{2} b^{2}-2 b^{2} c^{2}-2 c^{2} a^{2}=2 a^{2} b^{2}

\left(a^{2}+b^{2}-c^{2}\right)^{2}=2 a^{2} b^{2}

\left(a^{2}+b^{2}-c^{2}\right)^{2}=\frac{1}{2} \times {4} a^{2} b^{2}

\frac{\left(a^{2}+b^{2}-c^{2}\right)^{2}}{4 a^{2} b^{2}}=\frac{1}{2}

\left[\frac{a^{2}+b^{2}-c^{2}}{2 a b}\right]^{2}=\frac{1}{2}

\cos ^{2} C =\frac{1}{2}

\cos C=\pm \sqrt{\frac{1}{2}}


\cos C=\pm \frac{1}{\sqrt{2}}

CASE-I
\cos C=\frac{1}{\sqrt{2}}

cos C=cos 45°


c=45°


CASE-II

\cos C=-\frac{1}{\sqrt{2}}

cos C=-cos45°

cos C=cos(180°-45°)

cos C=cos 135°

C=135°


Question 15

Sol :
\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}=k

b+c=11k , c+a=12k , a+b=13k

2a+2b+2c=11k+12k+13k

2(a+b+c)=36k

a+b+c=18k

a=7k , b=6k , c=5k

\frac{\cos A}{7}=\frac{c^{2}+b^{2}-a^{2}}{2 b c \cdot 7}

=\frac{(5 k)^{2}+(6 k)^{2}-(7 k)^{2}}{2 \times 6 k \times 5 k \times 7}

=\frac{25 k^{2}+36 k^{2}-49 k^{2}}{2 \times 6 k \times 5 k \times 7}

=\frac{12 k^{2}}{12 k^{2} \times 35}=\frac{1}{35}

\frac{\cos B}{19}=\frac{c^{2}+a^{2}-b^{2}}{2ca\times 19}

=\frac{(5 k)^{2}+(7 k)^{2}-(6 k)^{2}}{2 \times 5 k \times 7 k \times 19}

=\frac{25 k^{2}+49 k^{2}-36 k^{2}}{38 k^{2} \times 35}

=\frac{38 k^{2}}{38 k^{2}+35}=\frac{1}{35}

\frac{\cos C}{25}=\frac{a^{2}+b^{2}-c^{2}}{2 a b \times 25}

=\frac{(7 k)^{2}+(6 k)^{2}-(5 k)^{2}}{2 \times 7 k \times 6 k \times 25}

=\frac{49 k^{2}+36 k^{2}-25 k^{2}}{2 \times 7 k \times 6 k \times 25}

=\frac{60 k^{2}}{2 \times 7 \times 6 \times k^{2} \times 25}=\frac{1}{35}

\therefore \frac{\cos A}{7}=\frac{cos B}{19}=\frac{cos C}{25}

Question 16

Sol :
∵A+B+C=180°

2B+B=180°

3B=180°

B=\dfrac{180^{\circ}}{3}=60^{\circ}

\cos B=\frac{c^{2}+a^{2}-b^{2}}{2 c a}

\cos 60^{\circ}=\frac{c^{2}+a^{2}-b^{2}}{2 c a}

\frac{1}{2}=\frac{c^{2}+a^{2}-b^{2}}{2 c a}

c^{2}+a^{2}-b^{2}=c a

a^{2}-c a+c^{2}=b^{2}

R.H.S

\frac{a+c}{\sqrt{a^{2}-a c+c^{2}}}=\frac{a+c}{\sqrt{b^{2}}}

=\frac{a+c}{b}

[∵ we know that ]

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB , c=ksinC

Now , \frac{a+c}{b}=\frac{k \sin A+k \sin }{k\sin B}

=\frac{k( \sin A+\sin C)}{K \sin B}

\frac{a+c}{\sqrt{a^{2}-a c+c^{2}}}=\frac{2 \sin\frac{A+C}{2} \cos \frac{A-C}{2}}{\sin B}

=\frac{2 \sin \left(\frac{180^{\circ}-B}{2}\right) \cos \frac{A-C}{2}}{\sin B}

\frac{a+c}{\sqrt{a^{2}-a c+c^{2}}}=\frac{2 \sin \left(\frac{180^{\circ}}{2}-\frac{B}{2}\right) \cos \frac{A-C}{2}}{\sin B}

=\frac{2 \cos \frac{B}{2} \cos \frac{A-C}{2}}{2+\frac{B}{2} \cos \frac{B}{2}}

\frac{a+c}{\sqrt{a^{2}-a c+c^{2}}}=\frac{\cos \frac{A-C}{2}}{\sin \frac{60^{\circ}}{2}}

\frac{a+c}{\sqrt{a^{2}-a c+c^{2}}}=\frac{\cos \frac{A-C}{2}}{\frac{1}{2}}

\frac{a+c}{\sqrt{a^{2}-a c+c^{2}}}=2 \cos \frac{A-C}{2}


Question 17

Sol :

a+b=√3(a-b)

\frac{1}{\sqrt{3}}=\frac{a-b}{a+b}

\tan \frac{A-B}{2}=\frac{a-b}{a+b} \cdot \cot \frac{c}{2}

\tan \frac{A-B}{2}=\frac{1}{\sqrt{3}} \times \cot \frac{60^{\circ}}{2}

\tan \frac{A-B}{2}=\frac{1}{\sqrt3} \times \sqrt{3}

\frac{\tan A-B}{2}=\tan 45^{\circ}

\frac{A-B}{2}=45^{\circ}

⇒A-B=90°

Question 18

Sol :
(i)
L.H.S
2\left(a \sin ^{2} \frac{C}{2}+\operatorname{csin}^{2} \frac{A}{2}\right)=c+a-b

=2\left[a \times \frac{(s-a)(s-b)}{a b}+c \frac{(s-b)(s-c)}{b c}\right]

=2\left[\frac{(s-a)(s-b)+(s-b)(s-c)}{b}\right]

=2\left[\frac{(s-b)[s-a+s-c)}{b}\right]

=2\left[\frac{(s-b)(2 s-a-c)}{b}\right]

=2\left[\frac{(s-b)\left(a+b+c-a-c\right)}{b}\right]

=\frac{2(s-b) \cdot b}{b}

=a+b+c-2b

=a+c-b


(ii) b \cos ^{2} \frac{C}{2}+c \cos ^{2} \frac{B}{2}=\frac{1}{2}(a+b+c)
Sol :
L.H.S
b \cos ^{2} \frac{C}{2}+c \cos ^{2} \frac{B}{2}

={b \cdot \frac{s(s-c)}{a b}}+c \cdot \frac{s(s-b)}{a c}

=\frac{s(s-c)+s(s-b)}{a}

=\frac{S(s-c+s-b)}{a}

=\frac{s(2s-c-b)}{a}

=\frac{S\left(a+b+ c-c-b\right)}{a}

=\frac{S \times a}{a}

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


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

Sol :
L.H.S
\frac{\cos A}{\operatorname{ccos} B+b \cos C}+\frac{\cos B}{a \cos C+\cos A}+\frac{\cos C}{b \cos A+a \cos B}

=\frac{\frac{b^{2}+c^{2}-a^{2}}{2 b c}}{\frac{a}{1}}+\frac{\frac{a^{2}+c^{2}-b^{2}}{2 a c}}{\frac{b}{1}}+\frac{\frac{a^{2}+b^{2}-c^{2}}{2 a b}}{\frac{c}{1}}

=\frac{b^{2}+c^{2}-a^{2}}{2 a b c}+\frac{a^{2}+c^{2}-b^{2}}{2 a b c}+\frac{a^{2}+b^{2}-c^{2}}{2 a b c}

=\frac{b^{2}+c^{2}-a^{2}+a^{2}+c^{2}-b^{2}+a^{2}+b^{2}-c^{2}}{2 a b c}

=\frac{a^{2}+b^{2}+c^{2}}{2 a b c}

Question 19

(i) a(\cos B+\cos C-1)+b(\cos C+\cos A-1)+c(\cos A+\cos B-1)=0
Sol :
L.H.S
=a(\cos B+\cos C-1)+b(\cos C+\cos A-1)+c(\cos A+\cos B-1)

=acosB+acosC-a+bcosc+bcosA-b+ccosA+ccosB-c

=(acosB+bcosA)+(acosC+ccosA)+(bcosC+ccosB)-a-b-c

=c++b+a-a-b-c

=0

(ii) acos(A+B+C)-bcos(B+A)-ccos(A+C)=0
Sol :
L.H.S
=acos(A+B+C)-bcos(B+A)-ccos(A+C)

=acos180°-bcos(180°-C)-ccos(180°-B)

=a(-1)-b(-cosC)-c(-cosB)

=a(-1)-bcosC+ccosB

=-a+a

=0


Question 20

(i) \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
Sol :

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB , c=ksinC

\left(b^{2}-c^{2}\right) cot A=\left(k^{2} \sin^{2} B+k^2 \sin ^{2} C\right)\cot A

=k^{2}\left(\sin ^{2} B-\sin ^{2} C\right) \cot A

=k^{2} \sin (B+c) \sin (B-c) \frac{\cos A}{\sin A}

=k^{2} \sin (B+c) \sin (B-C) \cdot \sin(B-C).\dfrac{\cos A}{\sin(180^{\circ}-(B+C))}

=k^{2} \sin (B+C) \sin (B-C) \frac{\cos A}{\sin (B+C)}

=k^{2}[\sin B \cos C-\cos B \sin C] \cos A

\left(b^{2}-c^{2}\right) \cot A=k^{2}\left[\cos A \sin B \cos C-\cos A \cos B \sin C\right]

\left(c^{2}-a^{2}\right) \cot B=k^{2} \sin (C-A) \cos B

=k^{2}(\sin C \cos A-\cos C \sin A) \cos B

\left(c^{2}-a^{2}\right) \cot B=k^{2}\left(\cos A \cos B \sin C-\sin A \cos B \cos C\right)..(ii)

\left(a^{2}-b^{2}\right) cot C=k^{2} \sin (A-B) \cos C

=k^{2}[\sin A \cos B-\cos A\sin B] \cos c

\left(a^{2}-b^{2}\right)\cot=k^{2}\left[\sin A \cos B \cos C -\cos A \sin B \cos C\right]..(iii)

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

=k^{2} \times 0

=0


(iii) \frac{1+\cos (A-B) \cos C}{1+\cos (A-C) \cos B}=\frac{a^{2}+b^{2}}{a^{2}+c^{2}}
Sol :
=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

L.H.S
\frac{1+\cos (A-B) \cos C}{1+\cos (A-C) \cos B}

=\frac{1+\cos (A-B) \cos \left(180^{\circ}-(A+B)\right.}{1+\cos (A-C) \cos \left[180^{\circ}-(A+C)\right]}

=\frac{1+\cos (A-B)\left[-\cos (A+B)\right]}{1+\cos(A-C)[-\cos (A+C)]}

=\frac{1-\cos (A-B) \cos (A+B)}{1-\cos(A-C) \cos (A+C)}

=\frac{1-\left(\cos ^{2} A-\sin ^{2} B\right)}{1-\left(\cos ^{2} A-\sin ^{2} C\right)}

=\frac{1-\cos^{2} A+\sin ^{2} B}{1-\cos ^{2} A+\sin ^{2} C}

=\frac{\sin ^{2} \theta+\sin ^{2} B}{\sin^{2} A+\sin ^{2} C}

=\frac{\frac{a^{2}}{k^{2}}+\frac{b^{2}}{k^{2}}}{\frac{a^{2}}{k^{2}}+\frac{c^{2}}{k^{2}}}

=\frac{\frac{a^{2}+b^{3}}{k^{2}}}{\frac{a^{2}+c^{2}}{k^{2}}}

=\frac{a^{2}+b^{2}}{a^{2}+c^{2}}


(iv) \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
Sol :
=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

\frac{b^{2}-c^{2}}{\cos B+\cos c}=\frac{k^{2} \sin ^{2} B-k^{2} \sin ^{2} C}{\cos B+\cos C}

=\frac{k^{2}\left[\sin ^{2} B-\sin ^2 C\right]}{\cos B+\cos C}

=k^{2} \frac{\left[\left(1-\cos ^{2} B\right)-\left(1-\cos ^{2}C\right)\right.}{\cos B+\cos c}

=\frac{k^{2}\left[1-\cos ^{2} B-1+\cos ^{2} C\right]}{\cos B+\cos C}

=\frac{k^{2}\left[\cos ^{2} C-\cos ^{2} B\right]}{\cos B+\cos C}

=\frac{k^{2}(\cos C+\cos B)(\cos C-\cos B)}{\cos B+\cos C}

\frac{b^{2}-c^{2}}{\cos B+\cos C}=k^{2}(\cos C-\cos B)..(i)

\frac{c^{2}-a^{2}}{\cos C+\cos A}=k^{2}(\cos A-\cos C)..(ii)

\frac{a^{2}-b^{2}}{\cos A+ \cos B}=k^{2}[\cos B-\cos A]..(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}=k^{2} \times 0=0


(v) \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
Sol :

=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

\frac{a^{2} \sin (B-C)}{\sin B+\sin C}=\frac{k^{2} \sin ^{2} A \sin (B-C)}{\sin B+\sin C}

=\frac{k^{2} \sin A \sin A \sin (B-C)}{\sin B+\sin C}

=\frac{k^{2} \sin A \sin [180^{\circ}-(B+C)] \sin (B \cdot C)}{\sin B+\sin C}

=\frac{k^{2} \sin A \sin (B+C) \sin (B-C)}{\sin B+\sin C}

=\frac{k^{2} \sin A\left(\sin ^{2} \theta-\sin ^{2} c\right)}{\sin B+\sin C}

=\frac{k^{2} \sin A(\sin B+\sin C)(\sin B-\sin C)}{\sin B+\sin C}

\frac{a^{2} \sin (B-C)}{\sin B+\sin C}=k^{2} \sin A(\sin B-\sin C)..(i)

\frac{b^{2} \sin (C-A)}{\sin C+\sin A}=k^{2} \sin B(\sin C-\sin A)..(ii)

\frac{c^{2} \sin (A-B)}{\sin A+\sin B}=k^{2} \sin C(\sin A-\sin B)..(iii)

\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}

=k^{2}[\sin A \sin B-\sin A \sin C+\sin B \sin C-\sin A +\sin A \sin C-\sin B \sin C]

=k^{2} \times 0=0


(vi) \frac{b^{2}-c^{2}}{a^{2}}=\frac{\sin (B-C)}{\sin (B+C)}
Sol :
=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

L.H.S
\frac{b^{2}-c^{2}}{a^{2}}=\frac{k^{2}\sin^{2} B^{2}-k^{2} \sin ^{2} C}{k^{2} \sin ^{2} A}

=\frac{k^{2}\left(\sin ^{2} B-\sin ^{2} c\right)}{k^{2} \sin ^{2}[180^{\circ}-(B+c)]}

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

=\frac{\sin (B-C)}{\sin (B+C)}


(vii) \tan \left(\frac{\mathrm{A}}{2}+\mathrm{B}\right)=\frac{c+b}{c-b} \tan \frac{\mathrm{A}}{2}
Sol :
R.H.S
\frac{c+b}{c-b} \tan \frac{A}{2}=\frac{1}{\frac{c-b}{c+b} \cot \frac{A}{2}}

=\frac{1}{\tan \frac{C-B}{2}}

=\cot \left(\frac{C-B}{2}\right)

=\cot \left(\frac{C}{2}-\frac{B}{2}\right)

[∵ A+B+C=180°
C=180°-A-B
=180°-(A+B)]

=\cot \left(\frac{180^{\circ}-A-B}{2}-\frac{B}{2}\right)

=\cot \left(\frac{180^{\circ}}{2}-\frac{A}{2}-\frac{B}{2}-\frac{B}{2}\right)

=\cot\left(90^{\circ}-\frac{A}{2}-B\right)

=\cot \left[90^{\circ}-\left(\frac{A}{2}+B\right)\right]

=\tan \left(\frac{A}{2}+B\right)

(viii)
Sol :
=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

b+c=2 a \cos \frac{B-C}{2}

\frac{b+c}{2 a}=\cos \frac{B-C}{2}

\frac{k \sin B+k \sin C}{2 k \sin A}=\cos \frac{B-C}{2}

\frac{k(\sin B+\sin C)}{2 k \sin A}=\cos \frac{B-C}{2}

\frac{2 \sin \frac{B+ C}{2} \cos \frac{B-C}{2}}{2 \sin A}=\cos \frac{B-C}{2}

\sin \frac{B+C}{2}=\sin A

\frac{B+C}{2}=A

B+C=2A

∴ A+B+C=180°

A+2A=180°

3A=180°

A=\dfrac{180^{\circ}}{3}=60^{\circ}


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

R.H.S
\frac{\cot \frac{c}{2}}{\tan \frac{A}{2}+\tan \frac{B}{2}}=\frac{\frac{s(s-c)}{\Delta}}{\frac{(s-b)(s-c)}{\Delta}+\frac{(s-c)(s-a)}{\Delta}}

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

=\frac{s(s-c)}{(s-c)(s-b+s-a)}

=\frac{s}{2 s-b-a}

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

=\frac{a+b+c}{2 c}


Question 21

(i) a \cos \frac{B-C}{2}=(b+c) \sin \frac{A}{2}
Sol :
\frac{a}{b+c}=\frac{\sin \cdot \frac{A}{2}}{\cos \frac{b-c}{2}}

=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

L.H.S
\frac{a}{b+c}=\frac{k \sin A}{k\sin B+k \sin C}

=\frac{k \sin A}{k(\sin B+\sin C)}

=\frac{\sin [180^{\circ}-(B+C)]}{\sin B+\sin C}

=\frac{\sin (B+C)}{\sin B+\sin C}

=\frac{2 \sin \frac{B+C}{2}}{2\sin\frac{B+C}{2} \cos \frac{B-C}{2}}

=\frac{\cos \left(\frac{180^{\circ}-A}{2}\right)}{\cos \frac{B-C}{2}}

=\frac{\cos \left(\frac{180^{\circ}}{2}-\frac{A}{2}\right)}{\cos \frac{B-C}{2}}

=\frac{\sin \frac{A}{2}}{\cos \frac{B-C}{2}}


(ii) \operatorname{asin}\left(\frac{A}{2}+B\right)=(b+c) \sin \frac{A}{2}
Sol :
\frac{a}{b+c}=\frac{\sin A}{\sin \left(\frac{A}{2}+B\right)}

\frac{a}{b+c}=\frac{k \sin A}{k\sin B+k \sin C}

=\frac{k \sin A}{k(\sin B+\sin C)}

L.H.S
\frac{a}{b+c}=\frac{k \sin A}{k \sin B+k \sin C}

=\frac{k \sin A}{k(\sin B+\sin C)}

=\frac{\sin [180^{\circ}-(B+C)]}{\sin B+\sin C}

=\frac{\sin (B+C)}{\sin B+\sin C}

=\frac{2 \sin \frac{B+C}{2} \cos \frac{B+C}{2}}{2 \sin \frac{B+C}{2} \cos \frac{B-C}{2}}

=\frac{\cos \left(\frac{180^{\circ}-A}{2}\right)}{\cos \left(\frac{B-(180^{\circ}-A-B)}{2}\right)}

=\frac{\cos \left(\frac{\sin }{2}-\frac{A}{2}\right)}{\cos \left(\frac{B-180^{\circ} +A+B}{2}\right)}

=\frac{\sin \frac{A}{2}}{\cos \left(\frac{-180^{\circ}+A+2 B}{2}\right)}

=\frac{\sin \frac{A}{2}}{\cos \left(-\frac{180^{\circ}-A-2 B}{2}\right)}

=\frac{\sin \frac{A}{2}}{\cos \left(\frac{180^{\circ}-A-2 B}{2}\right)}

=\frac{\sin \frac{A}{2}}{\cos \left( \frac{180}{2}-\frac{A}{2}-\frac{2 B}{2} \right)}

=\frac{\sin \frac{A}{2}}{\cos \left[90^{\circ}-\left(\frac{A}{2}+B\right)\right]}

=\frac{\sin \frac{A}{2}}{\sin \left(\frac{A}{2}+B\right)}


Question 22

(i) 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
Sol :
=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

L.H.S
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)

=k^{2} \sin ^{2} A\left(\cos ^{2} B-\cos ^{2} C\right)+k^{2} \sin ^{2} B\left(\cos ^{2}C\left.-\cos ^{2} A\right)+k^{2} \sin ^{2} C\left(\cos ^{2} A-\cos ^{2} B\right)\right.

=k^{2}\left(1-\cos ^{2} A\right)\left(\cos ^{2} B-\cos ^{2} C\right)+k^{2}\left(1-\cos ^{2} B\right)\left(\cos ^{2} C-\cos ^{2} A\right)+k^{2}\left(1-\cos ^{2} C\right)\left(\cos ^{2} A-\cos ^{2} B\right)

=k^{2}\left[\cos ^{2} B-\cos ^{2} C-\cos ^{2} A \cos ^{2} B+\cos ^{2} A \cos ^{2} C+\cos ^{2} C-\cos ^{2} A\right. -\cos ^{2} B \cos ^{2} C+\cos ^{2} B \cos ^{2} A+\cos ^{2} A-\cos ^{2} B-\cos ^{2} C \cos ^{2} A \left.+\cos ^{2} C \cos ^{2} B \right]

=k^{2} \times 0=0


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

R.H.S
\frac{\cos \frac{B}{2} \cos \frac{C}{2}}{\sin \frac{A}{2}}=\frac{\sqrt{\frac{s(s-b)}{a c}} \times \sqrt{\frac{S(s-c)}{a b}}}{\sqrt{\frac{(s-b)(s-c)}{b c}}}

=\frac{s}{a}=\frac{a+b+c}{2 a}

Question 23

Sol :
A=2B (given)

sinA=sin2B

sinA=2sinBcosB

Let =\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

\frac{a}{k}=2 \frac{b}{k} \times \frac{a^{2}+c^{2}-b^{2}}{2 a c}

a^{2} c=a^{2} b+b c^{2}-b^{3}

0=a^{2} b-a^{2} c-b^{3}+b c^{2}

0=a^{2}(b-c)-b\left(b^{2}-c^{2}\right)

0=a^{2}(b-c)-b(b-c)(b+c)

0=(b-c)\left[a^{2}-b(b+c)\right]

b-c=0
b=c

a^{2}-b(b+c)=0
a^{2}=b(b+c)


Question 24

\frac{\cos A}{a}+\frac{a}{b c}=\frac{\cos B}{b}+\frac{b}{c a}=\frac{\cos C}{c}+\frac{c}{a b}
Sol :
\frac{\cos A}{a}+\frac{a}{b c}=\frac{b^{2}+c^{2}-a^{2}}{2 a b c}+\frac{a}{b c}

=\frac{b^{2}+c^{2}-a^{2}+2 a^{2}}{2 a b c}

=\frac{a^{2}+b^{2}+c^{2}}{2 a b c}

\frac{\cos B}{b}+\frac{b}{c a}=\frac{a^{2}+c^{2}-b^{2}}{2 a b c}+\frac{b}{c a}

=\frac{a^{2}+c^{2}-b^{2}+2 b^{2}}{2 a b c}

=\frac{a^{2}+b^{2}+c^{2}}{2 a b c}

\frac{\cos C}{c}+\frac{c}{a b}=\frac{a^{2}+b^{2}-c^{2}}{2 a b c}+\frac{c}{a b}

=\frac{a^{2}+b^{2}-c^{2}+2 c^{2}}{2 a b c}

=\frac{a^{2}+b^{2}+c^{2}}{2 a b c}

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


Question 25

(i) \frac{\cos 2 \mathrm{A}}{a^{2}}-\frac{\cos 2 \mathrm{B}}{b^{2}}=\frac{1}{a^{2}}-\frac{1}{b^{2}}
Sol :
=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

L.H.S
\frac{\cos 2 A}{a^{2}}-\frac{\cos 2 B}{b^{2}}

=\frac{\cos 2 A}{k^{2} \sin ^{2} A}-\frac{\cos 2 B}{k^{2} \sin ^{2} B}

=\frac{\left(1-2 \sin ^{2} A\right)}{k^{2} \sin ^{2} A}-\frac{\left(1-2 \sin ^{2} B\right)}{k^{2} \sin^2 B}

=\left[\frac{1}{k^{2} \sin ^{2} A}-\frac{2 \sin ^{2} A}{k^{2} \sin ^{2} A}\right)-\left(\frac{1}{k^{2} \sin ^{2} B}-\frac{2 \sin ^{2} B}{k^{2} \sin ^{2} B}\right)

=\frac{1}{(k+\sin A)^{2}}-\frac{2}{k^{2}}-\frac{1}{(k \sin B)^{2}}+\frac{2}{k^{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
Sol :
=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

L.H.S
\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

\left(k^{2} \sin ^{2} B-k^{2} \sin ^{2} C\right) \sin ^{2} A +( k^{2} \sin ^{2}C-k^2 \sin ^{2} A) \sin ^{2} B+\left(k^{2} \sin ^{2} A-k^{2} \sin ^{2} B\right) \sin ^{2} C

\left.=k^{2}\left[\sin ^{2} B \sin ^{2} A-\sin ^{2} C \sin ^{2} A+\sin ^{2} C \sin ^{2} B-\sin ^{2} A \sin ^{2} B\right. +\sin ^{2} A \sin ^{2} C-\sin ^{2} B \sin ^{2} C\right]

=k^{2} \times 0=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
Sol :
=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

\frac{a^{2} \sin (B-C)}{\sin A}=\frac{k^{2} \sin ^{2} A \sin (B-C)}{\sin A}

=k^{2} \sin [180^{\circ} -(B+C)] \sin (B-C)

=k^{2} \sin (B+C) \sin (B-C)

\frac{a^{2}-\sin (B-C)}{\sin A}=k^{2}\left[\sin ^{2} B-\sin ^{2} C\right]..(i)

b^{2} \frac{\sin (C-A)}{\sin B}=k^{2}\left[\sin ^{2} C-\sin ^{2} A\right]..(ii)

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

=k^{2}\left[\sin ^{2} B-\sin ^{2} C+\sin ^{2} C-\sin ^{2} A+\sin^2 A- \sin^2 B\right]

=k^{2} \times 0

Question 26

Sol :
AD is median in ΔABC

BD=CD

In ΔABD
\sin B=\frac{A D}{B D} \Rightarrow A D=B D \sin B

In ΔADC
\frac{C D}{\sin (A-90^{\circ})}=\frac{A D}{\sin C}

\frac{B D}{-\sin \left(\sin 90^{\circ}-A\right)}=\frac{B D\sin B}{\sin C}

\frac{1}{-\cos A}=\frac{\sin B}{\sin C}

sinC=-cosAsinB

sinC+cosAsinB=0

sin[180°-(A+B)]+cosAsinB=0

sin(A+B)+cosAsinB=0

sinAcosB+cosAsinB+cosAsinB=0

sinAcosB+2cosAsinB=0

\frac{\sin A \cos B}{\cos A \cos B}+\frac{2 \cos A \sin B}{\cos A \cos B}=0

tanA+2tanB=0


Question 27

(i)
Sol :
\tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2}.. are in AP

\tan \frac{B}{2}-\tan \frac{A}{2}=\tan \frac{C}{2}-\tan \frac{B}{2}

\frac{\sin \frac{B}{2}}{\cos \frac{B}{2}}-\frac{\sin \frac{A}{2}}{\cos \frac{A}{2}}=\frac{\sin \frac{C}{2}}{\cos \frac{C}{2}}-\frac{\sin \frac{B}{2}}{\cos \frac{B}{2}}

\frac{\sin \frac{B}{2} \cos \frac{B}{2}-\cos \frac{B}{2} \sin\frac{A}{2}}{\cos \frac{B}{2} \cos \frac{A}{2}}=\frac{\sin \frac{C}{2} \cos \frac{B}{2}-\cos \frac{C}{2} \sin \frac{B}{2}}{\cos \frac{C}{2} \cos \frac{B}{2}}

\frac{\sin \left(\frac{B}{2}-\frac{A}{2}\right)}{\cos \frac{C}{2}}=\frac{\sin \left(\frac{C}{2}-\frac{B}{2}\right)}{\cos \frac{C}{2}}

\frac{-\sin \left(\frac{A}{2}-\frac{B}{2}\right)}{\cos\left[90^{\circ}-\left(\frac{B}{2}+\frac{C}{2}\right)\right]}=\frac{-\sin \left(\frac{B}{2}-\frac{C}{2}\right)}{\cos \left[90^{\circ}-\left(\frac{A}{2}+\frac{B}{2}\right)\right]}

\frac{\sin \left(\frac{A-B}{2}\right)}{\sin \left(\frac{B+C}{2}\right)}=\frac{\sin \left(\frac{B-C}{2}\right)}{\sin \left(\frac{A+B}{2}\right)}

2 \sin \left(\frac{A-B}{2}\right) \sin \left(\frac{A+B}{2}\right)=2 \sin \left(\frac{B-C}{2}\right) \sin \left(\frac{B+C}{2}\right)

cosB-cosA=cosC-cosB

∴ cosA, cosB , cosC are in A.P


(ii)
Sol :
=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA , b=ksinB  , c=ksinC

\frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)}

sinAsin(B-C)=sinCsin(A-B)

\sin [180^{\circ}-(B+C)] \sin (B-C)=\sin [180^{\circ}-(A+B)] \sin (A-B)

sin(B+C)sin(B-C)=sin(A+B)sin(A-B)

\sin ^{2} B-\sin ^{2} C=\sin ^{2} A-\sin ^{2} B

\frac{b^{2}}{k^{2}}-\frac{c^{2}}{k^{2}}=\frac{a^{2}}{k^{2}}-\frac{b^{2}}{k^{2}}

\frac{b^{2}-c^{2}}{k^{2}}=\frac{a^{2}-b^{2}}{k^{2}}

-\left(c^{2}-b^{2}\right)=-\left(b^{2}-a^{2}\right)

a^{2}, b^{2}, c^{2} are in A.P


(iii)
Sol :
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=\frac{1}{k}

ak=sinA , bk=sinB , ck=sinC

∵ sinA , sinB , sinC are in A.P

sinB-sinA=sinC-sinB

bk-ak=ck-bk

k(b-a)=k(c-b)

b+b=a+c

2b=a+c

2b+b=a+b+c

3b=2s

3b+s=2s+s

3b+s=3s

s=3s-3b

s=3(s-b)

\frac{1}{3}=\frac{s-b}{s}

\frac{1}{3}=\tan \frac{A}{2} \tan \frac{C}{2}

1=3 \tan \frac{A}{2} \tan \frac{C}{2}

Question 28

\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=\cos B \cos C+\cos C \cdot \cos A+\cos A \cos B
Sol :

2 \cos ^{2} A+2 \cos ^{2} B-2 \cos ^{2} C=2 \cos B \cos C+2 \cos C \cos A+2 \cos A \cos B

\cos^{2} a+\cos^{2} A+\cos^{2} B+\cos^{2} B+\cos^{2} C+\cos^{2} C-2 \cos B \cos C-2\cos C \cos A-2\cos A \cos B=0

(cos2B+cos2C-2cosBcosC)+(cos2C+cos2A-2cosCcosA)+(cos2A+cos2B-2cosAcosB)=0

(cosB-cosC)2+(cosC-cosA)2+(cosA-cosB)2=0

cosB-cosC=0
cosB=cosC
B=C

cosC-cosA=0
cosC=cosA
C=A

cosA-cosB=0
cosA=cosB
A=B

∴A=B=C


Question 30

Sol :
\sin C=\frac{\sin A+\sin B}{\cos A+\cos B}

\sin [180^{\circ}-(A+B)]=\frac{\sin A+\sin B}{\cos A+\cos B}

\sin (A+B)=\frac{2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}}{2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}}

2+ \frac{A+B}{2} \cos \frac{A+B}{2=\frac{\sin \frac{A+B}{2}}{\cos \frac{A+B}{2}}

\cos ^{2} \frac{A+B}{2}=\frac{1}{2}

\cos ^{2} \frac{A+B}{2}=\left(\frac{1}{\sqrt{2}}\right)^{2}

\cos ^{2} \frac{A+B}{2}=\cos ^{2} 45^{\circ}

\frac{A+B}{2}=45^{\circ}

A+B=90°


A+B+C=180°
90°+C=180°
C=90°


Question 31

Sol :
\frac{\cos A}{b}=\frac{\cos B}{a}

acosA=bcosB

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k

a=ksinA, b=ksinB, c=ksinC


ksinAcosA=ksinBcosB

2sinAcosA=2sinBcosB

sin2A=sin2B

sin2A-sin2B=0

2 \cos \frac{2A+2 B}{2} \sin \frac{2 A-2 B}{2}=0

cos(A+B)sin(A-B)=0


cos(A+B)=0
cos(A+B)=cos90°
A+B=90°


sin(A-B)=0
sin(A-B)=sin0°
A-B=0
A=B

A+B+C=180°
90°+C=180°
C=90°


Question 34

(i) b \cos ^{2} \frac{C}{2}+\operatorname{ccos}^{2} \frac{B}{2}=\operatorname{ccos}^{2} \frac{A}{2}+a \cos ^{2} \frac{C}{2}=a \cos ^{2} \frac{B}{2}+b \cos ^{2} \frac{A}{2}=\frac{a+b+c}{2}
Sol :
b \cos ^{2} \frac{c}{2}+c \cos ^{2} \frac{B}{2}=b(\sqrt{\frac{s(s-c)}{a b}})^{2}+c(\sqrt{\frac{s(s-b)}{a c}})^{2}

=\frac{b s(s-c)}{a b}+\frac{c s(s-b)}{ac}

=\frac{s(s-c)+s(s-b)}{a}

=\frac{a[s-c+s-b]}{a}

=\frac{s[2s-c-b]}{a}

=\frac{5[a+b+c-c-b]}{a}

=\frac{5 \times a}{a}=s=\frac{a+b+c}{2}

c \cos ^{2} \frac{A}{2}+a \cos ^{2} \frac{C}{2}=C\left.\left(\sqrt{\left.\frac{s(s-a)}{b c}\right.}\right)^{2}+a\left(\sqrt{\frac{s(s-c)}{a b}}\right)^{2}\right.

=\frac{c \times s(s-a)}{bc}+a\times \frac{s(s-c)}{a b}

=\frac{s(s-a)+s(s-c)}{b}

=\frac{s[s-a+s-c]}{b}

=\frac{s[2s-a-c]}{b}

=\frac{s(a+b+c-a-c)}{b}

=\frac{s \times b}{b}=s

=\frac{a+b+c}{2}


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

=a\frac{s(s-b)}{a c}+b \times \frac{s(s-a)}{b c}

=\frac{s(s-b)+s(s-a)}{c}

=\frac{s[s-b+s-a]}{c}

=\frac{s[2s-b-a]}{c}

=\frac{s(a+b+c-b-a]}{c}

=\frac{s \times c}{c}=s=\frac{a+b+c}{2}


(ii) \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
Sol :
L.H.S
\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}

=\frac{b-c}{a} \times\left( \frac{s(s-a)}{b c}\right)^{2}+\frac{c-a}{b} \times\left(\sqrt{\frac{s(s-b)}{a c}}\right)^{2}+\frac{a-b}{c} \times\left(\sqrt{\frac{s(s-c)}{a b}}\right)^{2}

=\frac{(b-c) \cdot s(s-a)}{a b c}+\frac{(c-a) \cdot s(s-b)}{a b c}+\frac{(a-b) \cdot s(s-c)}{a b c}

=\frac{s}{a b c}[(b-c)(s-a)+(c-a)(s-b)+(a-b)(s-c)]

=\frac{S}{a b c}[5 b-a b-s c+c a+s c-b c-5 a+a b+s a-c q

=\frac{s}{a b c} \times 0=0


(iii) (b-c) \cot \frac{A}{2}+(c-a) \cot \frac{B}{2}+(a-b) \cot \frac{C}{2}=0
Sol :
L.H.S
(b-c) \cot \frac{A}{2}+(c-a) \cot \frac{B}{2}+(a-b) \cot \frac{C}{2}

=(b-c) \cdot \frac{s(s-a)}{\Delta}+(c-a) \frac{s(s-b)}{\Delta}+(a-b) \cdot \frac{s(s-c)}{\Delta}

=\frac{5}{\Delta}[(b-c)(s-a)+(c-a)(s-b)+(a-b)(s-c)]

=\frac{S}{\Delta} \times 0=0


Question 35

Sol :
a\cos ^{2} \frac{C}{2}+\operatorname{ccos}^{2} \frac{A}{2}=\frac{3 b}{2}

a\left(\sqrt{\frac{5(s-2)}{a b}}\right)^{2}+c\left(\sqrt{\frac{s(s-a)}{b c}}\right)^{2}=\frac{3 b}{3}

a \times \frac{s(s-c)}{a b}+c \times \frac{s(s-a)}{b c}=\frac{3 b}{2}

\frac{s(s-c)+s(s-a)}{b}=\frac{3 b}{2}

\frac{s[s-c+s-a]}{b}=\frac{3 b}{2}

\frac{s[2s-c-a]}{b}=\frac{3 b}{2}

\frac{s[a+b+c-c-a]}{b}=\frac{3 b}{2}

\frac{s \times b}{b}=\frac{3b}{2}

2s=3b

a+b+c=3b

a+c=2b

or c-b=b-a

∴ a ,b,c are in A.P


Question 36

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

R.H.S
\frac{\cot \frac{C}{2}}{\tan \frac{A}{2}+\tan \frac{B}{2}}=\frac{\frac{s(s-c)}{\Delta}}{\frac{(s-b)(s-c)}{\Delta}+\frac{(s-c)(s-a)}{\Delta}}

=\frac{\frac{s(s-c)}{\Delta}}{\frac{(s-b)(s-c)+(s-c)(s-q)}{\Delta}}

=\frac{s(s-c)}{(s-c)[s b+s-a]}

=\frac{s}{[2s-b-a]}

=\frac{5}{(a+b+c-b-a)}

=\frac{a+b+c}{2}


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

=1-\frac{(s-b)(s-c)}{s} \times \frac{(s-c)(s-a)}{\Delta}

=\frac{\Delta^{2}-(s-a)(s-b)(s-c)^{2}}{\Delta^{2}}

[∵\delta=\sqrt{s(s-a)(s-b)(s-c)}]

=\dfrac{s(s-a)(s-b)(s-c)-\frac{s(s-a)(s-b)(s-c)^{2}}{2}}{s(s-a)(s-b)(s-c)}

=\dfrac{s(s-a)(s-b)(s-c)\left[1-\frac{s(s-c)}{s}\right]}{s(s-a)(s-b)(s-c)}

=1-\frac{s}{s}+\frac{c}{s}

=1-1+\dfrac{C}{\frac{a+b+c}{2}}

=\dfrac{2 C}{a+b+c}


(iii) 1-\tan \frac{\mathrm{A}}{2} \tan \frac{\mathrm{B}}{2}=\frac{2 c}{a+b+c}
Sol :
L.H.S
1-\tan \frac{A}{2} \tan \frac{B}{2}

[∵ \tan \frac{A}{2}\tan\frac{B}{2}=\frac{s-c}{s}]

=1-\frac{s-c}{s}

=1-\frac{s}{s}+\frac{c}{s}

=1+1+\frac{c}{\frac{a+b+c}{2}}=\frac{2 c}{a+b+c}


Question 37

(i) (b+c-a) \tan \frac{A}{2}=(c+a-b) \tan \frac{B}{2}=(a+b-c) \tan \frac{C}{2}
Sol :
(b+c-a) \tan \frac{A}{2}=(a+b+c-2 a) \frac{(s-b)(s-c)}{\Delta}

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

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

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

=\dfrac{(2s-2 b)(s-c)(s-a)}{\Delta}

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

=2\frac{(s-a)(s-b)(s-c)}{2}


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

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

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


(ii) \frac{\cos ^{2} \frac{A}{2}}{a}+\frac{\cos ^{2} \frac{C}{2}}{b}+\frac{\cos ^{2} \frac{C}{2}}{c}=\frac{s^{2}}{a b c}
Sol :
L.H.S
\frac{\cos ^{2} \frac{A}{2}}{a}+\frac{\cos ^{2} \frac{B}{2}}{b}+\frac{\cos ^{2} \frac{C}{2}}{c}

=\dfrac{\left(\sqrt{\left.\frac{s(s-a)}{b c}\right.}\right)^2}{a}+\dfrac{\left(\sqrt{\frac{s(s-b)}{a c}}\right)^2}{b}+\dfrac{\left(\sqrt{\frac{s(s-c)}{a b}}\right)^{2}}{c}

=\frac{s(s-a)}{a b c}+\frac{s(s-b)}{a b c}+\frac{s(s-c)}{a b c}

=\frac{s(s-a)+s(s-b)+s(s-c)}{a b c}

=\frac{s[s-a+s-b+s-c]}{a b c}

=\frac{s[3 s-(a+b+c)]}{a b c}

=\frac{s[3 s-2 s]}{a b c}

=\frac{s \times s}{a b c}

=\frac{s^{2}}{a b c}


Question 38

(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)=\operatorname{ccot} \frac{C}{2}
Sol :
L.H.S
\left(\cot\frac{A}{2}+\cot\frac{B}{2}\right)\left(a\sin^2 \frac{B}{2}+b \sin ^{2} \frac{A}{2}\right)

=\left[\frac{s(s-a)}{\Delta}+\frac{s(s-b]}{\Delta}\right]\left[a\left(\sqrt{\frac{(s-c)(s-a)}{c a}}\right)^{2}+b\left(\sqrt{\frac{(s-b)(s-c)}{b c}}\right)^{2}\right]

=\frac{s}{\Delta}[s-a+s-b)\left[\begin{array}{cccc}a \frac{(s-c)(s-a)}{c a} +b\left(\frac{s-b)(s-c)}{b c}\right.\end{array}\right]

=\frac{s}{\Delta}(2s-a-b) \frac{(s-c)}{c}[s-a+s-b]

=\frac{S}{\Delta}\left(a+b+c-a-b\right) \frac{(s-c)}{c}(a+b+c-a-b)

=\frac{s}{\Delta} \times c \times \frac{(s-c)}{c} \times c

=c \cdot \frac{s(s-c)}{\Delta}=c \cdot \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)}
Sol :
L.H.S
\tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}

=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \times \sqrt{\frac{(s-c)(s-a)}{s(s-b)}} \times \sqrt{\frac{(s-a)(s-b)}{s(s-c)}}

=\sqrt{\left(\frac{s}{s}-\frac{c}{s}\right)} \times \sqrt{\left(\frac{s-a}{s}-\frac{a}{s}\right)} \times \sqrt{\left(\frac{s}{s}-\frac{b}{s}\right)}

=\sqrt{\left(1-\frac{a}{s}\right)\left(1-\frac{b}{s}\right)\left(1-\frac{c}{s}\right)}


Question 40

Sol :
[]
𝛼-β+𝛼+𝛼+β=3×side

3𝛼=3×side

𝛼=side


[]
\sqrt{s(s-a)(s-b)(s-c)}=\frac{3}{s} \times \frac{\sqrt{3}}{4}(side)^2

\sqrt{\frac{3 \alpha}{2}\left[\frac{3 \alpha}{2}-(\alpha-\beta)\right]\left[\frac{3 \alpha}{2}-\alpha\right]\left[\frac{3 \alpha}{2}-(\alpha+\beta)\right.}=\frac{3}{5} \times \frac{\sqrt{3}}{4} \alpha^{2}

\sqrt{\frac{3 \alpha}{2} \times \left(\frac{\alpha}{2}+\beta\right) \cdot \frac{\alpha}{2} \times\left(\frac{\alpha}{2}-\beta\right)}=\frac{3}{5} \times \frac{\sqrt{3}}{4} \alpha^{2}

\sqrt{\frac{3 \alpha}{2} \times\left(\frac{\alpha+2 \beta}{2}\right) \times \frac{\alpha}{2} \times\left(\frac{\alpha-2 \beta}{2}\right)}=\frac{3}{5} \times \frac{\sqrt{3}}{4} \alpha^{2}

\frac{\alpha}{2} \times \frac{1}{2} \sqrt{3(\alpha+2 \beta)(\alpha-2 \beta)}=\frac{3}{5} \times \frac{\sqrt{3}}{4} \alpha^{2}

\frac{\alpha}{4} \sqrt{3} \times \sqrt{(\alpha+2 \beta)(\alpha-2 \beta)}=\frac{3}{5} \times \frac{\sqrt{3}}{4} \alpha ^2

\sqrt{\alpha^{2}-(2 \beta)^{2}}=\frac{3}{5} \alpha

[]

(\sqrt{\alpha^{2}-4 \beta^{2}})^{2}=\left(\frac{3}{5} \alpha\right)^{2}

\alpha^{2}-4 \beta^{2}=\frac{9 \alpha^{2}}{25}

25 \alpha^{2}-100 \beta^{2}=9 \alpha^{2}

16 \alpha^{2}=100 \beta^{2}

\frac{\alpha^{2}}{\beta^{2}}=\frac{100}{16}

\frac{\alpha}{\beta}=\sqrt{\frac{25}{4}}=\frac{5}{2}

let 𝛼=5k ,β=2k




First side=𝛼-β=5k-2k=3k

second side=𝛼=5k

third side=𝛼+β=5k+2k=7k


ratios of sides 3:5:7



Question 41

Sol :
a=3m ,b=4m , c=5m , d=6m

A+C=120°

s=\frac{a+b+c+d}{2}=\frac{3+4+5+6}{2}

=\dfrac{18}{2}=9cm


=\sqrt{(s-a)(s-b)(s-c)(s-d)-a b c d \cos ^{2}\left(\frac{A+C}{2}\right)}

=\sqrt{(9-3)(9-4)(9-5)(9-6)-3 \times 4 \times 5 \times 6 \cos ^{2} \frac{120^{\circ}}{2}}

=\sqrt{6 \times 5 \times 4 \times 3-3 \times 4 \times 5 \times 6 \times\left(\frac{1}{2}\right)^{2}}

=\sqrt{3 \times 4 \times 5 \times 6\left(1-\frac{1}{4}\right)}

=\sqrt{3 \times 2 \times 2 \times 5 \times 3 \times 2 \times \frac{3}{4}}

=3 \sqrt{2 \times 5 \times 3}

=3 \sqrt{30} \mathrm{cm}^{2}



Question 42

r_{1}+r_{2}+r_{3}-r=4 R
Sol :
L.H.S
r_{1}+r_{2}+r_{3}-r

=\frac{\Delta}{s-a}+\frac{\Delta}{s-b}+\frac{\Delta}{s-c}-\frac{\Delta}{s}

\left.=\frac{\Delta[s(s-b)(s-c)+s(s-a)(s-c)+s(s-a)(s-b)-(s-a)(s-b)(s-c)}{s(s-a)(s-b)(s-c)}\right]

[∵ \Delta=\sqrt{s(s-a)(s-b)(s -c)} ]

=\frac{\Delta}{\Delta^{2}}\left[\begin{array}{c}s\left(s^{2}+s c-s b+b c\right)+s\left(s^{2}-s c-s a+a c\right)+s\left(s^{2}-s b-s a+a b\right) -(s-a)\left(s^{2}-s c-s b+b c\right)\end{array}\right]

\begin{aligned} {=\frac{1}{\Delta}\left[s^{3}-s^{2} c-s^{2} b+s b c+s^{3}-s^{2} c-s^{2} a+s a c+s^{3}-s^{2} b\right.}{-s^{2} a+s a b-s^{3}+s^{2} c+s^{2} b-s b c+s^{2} a-s a c}-s a b+a b c] \end{aligned}

=\frac{1}{\Delta}\left[2 s^{3}-s^{2} c-s^{2} b-s^{2} a+a b c\right]

=\frac{1}{\Delta}\left[2s^{3}-s^{2}(c+b+a)+a b c\right]

=\frac{1}{\Delta}\left[2 s^{3}-s^{2} \cdot 2 s+a b c\right]

=\frac{1}{\Delta}\left[2 s^{3}-2 s^{3}+a b c\right]

=\frac{a b c}{\Delta}=4R

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