Chapter 10 : Sine and Cosine Formulae and Their Applications - Rd Sharma Solutions for Class 11-science Maths CBSE

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Chapter 10 - Sine and Cosine Formulae and Their Applications Exercise Ex. 10.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

In any triangle ABC, prove the following:

b sinB – c sinC = a sin (B - C)

Solution 11

Question 12

In any triangle ABC, prove the following:

a2sin(B - C)= (b2 –c2)sinA

Solution 12

Question 13

Solution 13

Question 14

In any triangle ABC, prove the following:

a(sinB - sinC) + b (sinC - sinA) + c (sinA - sinB) = 0

Solution 14

Question 15

Solution 15

Question 16

In any triangle ABC, prove the following:

a2(cos2B – cos2C) + b2(cos2C – cos2A) + c2(cos2A –cos2B) = 0

Solution 16

Question 17

In any triangle ABC, prove the following:

b cosB + c cosC = a cos(B - C)

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

In any triangle ABC, prove the following:

a cosA + b cosB + c cosC= 2b sinA sinC= 2c sinA sinB

Solution 22

Question 23

a(cos B cosC + cosA)= b(cos C cosA + cosB)= c(cos A cosB + cosC)

Solution 23

Question 24

Solution 24

Question 25

In ΔABC prove that, if Ө be any angle, then b cosӨ = c cos(A - Ө) + a cos(C + Ө)

Solution 25

Question 26

In a ΔABC, if sin2A + sin2B = sin2C, show that the triangle is right angled.

Solution 26

Question 27

In any ΔABC, if a2, b2, c2 are in A.P., prove that cot A, cot B and cot C are also in A.P.

Solution 27

Question 28

The upper part of a broken over by the wind makes an angle of 300 with the ground and the distance from the root to the point where the top of the tree touches the ground is 15m. Using sine rule, find the height of the tree.

Solution 28

Question 29

At the foot of a mountain the elevation of its summit is 450; after ascending 1000m towards the mountain up a slope of 300 inclination, the elevation is found to be 600. Find the height of the mountain.

Solution 29

Question 30

A space p e r s o n space o b s e r v e s space t h e space a n g l e space o f space e l e v a t i o n space o f space t h e space p e a k space o f space a space h i l l space f r o m space a space
s t a t i o n space t o space b e space apostrophe alpha apostrophe. space H e space w a l k s space apostrophe c apostrophe space m e t r e s space a l o n g space a space s l o p e space i n c l i n e d space a t space t h e space a n g l e space apostrophe beta apostrophe
a n d space f i n d s space t h e space a n g l e space o f space e l e v a t i o n space o f space t h e space p e a k space o f space t h e space h i l l space t o space b e space apostrophe gamma apostrophe. space
S h o w space t h a t space t h e space h e i g h t space o f space t h e space p e a k space a b o v e space t h e space g r o u n d space i s space fraction numerator c cross times sin alpha cross times sin open parentheses gamma minus beta close parentheses over denominator sin open parentheses gamma minus alpha close parentheses end fraction.

Solution 30

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.

T h e space p e r s o n space i s space o b s e r v i n g space t h e space p e a k space P space f r o m space t h e space p o i n t space Q.
T h e space d i s tan c e space h e space t r a v e l l e d space i s space Q T equals c space m e t r e s space a n d space t h e space a n g l e space o f space i n c l i n a t i o n space o f
Q T space i s space beta.
H e space i s space o b s e r v i n g space t h e space p e a k space f r o m space t h e space p o i n t space a n d space t h e space a n g l e space o f space i n c l i n a t i o n space i s space gamma.
N o w space c o n s i d e r space t h e space t r i a n g l e space capital delta Q U T.
angle T Q U equals beta minus alpha
T h u s comma space sin open parentheses alpha minus beta close parentheses equals a over c
rightwards double arrow a equals c cross times sin open parentheses alpha minus beta close parentheses.... left parenthesis 1 right parenthesis
N o w space c o n s i d e r space t h e space t r i a n g l e space capital delta P Q R.
W e space k n o w space t h a t space angle Q P R equals 90 degree minus alpha
I n space t r i a n g l e space capital delta P T S comma space angle T P S equals 90 degree minus gamma
T h u s comma space angle T P U equals angle Q P R minus angle T P S
rightwards double arrow angle T P U equals open parentheses 90 degree minus alpha close parentheses minus open parentheses 90 degree minus gamma close parentheses
rightwards double arrow angle T P U equals gamma minus alpha
N o w space c o n s i d e r space t h e space capital delta T P U comma
T h u s comma space sin open parentheses gamma minus alpha close parentheses equals a over b
rightwards double arrow b equals fraction numerator a over denominator sin open parentheses gamma minus alpha close parentheses end fraction
S u b s t i t u t i n g space t h e space v a l u e space o f space a space f r o m space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma
b equals fraction numerator c cross times sin open parentheses alpha minus beta close parentheses over denominator sin open parentheses gamma minus alpha close parentheses end fraction... left parenthesis 2 right parenthesis
W e space n e e d space t o space f i n d space t h e space t o t a l space h e i g h t space o f space t h e space p e a k space P R.
H e r e comma space P R equals P S plus S R.... left parenthesis 3 right parenthesis
F r o m space t h e space t r i a n g l e space P S T comma space
sin gamma equals fraction numerator P S over denominator P T end fraction equals fraction numerator P S over denominator b end fraction
rightwards double arrow P S equals b sin gamma.... left parenthesis 4 right parenthesis
F r o m space t h e space t r i a n g l e space Q T W comma space
sin beta equals fraction numerator T W over denominator Q T end fraction equals fraction numerator T W over denominator c end fraction
rightwards double arrow T W equals S R equals c sin beta.... left parenthesis 5 right parenthesis
S u b s t i t u t i n g space t h e space v a l u e s space o f space P S space a n d space S R space f r o m space e q u a t i o n s space left parenthesis 4 right parenthesis space a n d space left parenthesis 5 right parenthesis
i n space e q u a t i o n space left parenthesis 3 right parenthesis comma space w e space h a v e
P R equals P S plus S R
rightwards double arrow P R equals b sin gamma plus c sin beta
rightwards double arrow P R equals fraction numerator c cross times sin open parentheses alpha minus beta close parentheses over denominator sin open parentheses gamma minus alpha close parentheses end fraction sin gamma plus c sin beta space space space space space space left square bracket f r o m space e q u a t i o n space left parenthesis 2 right parenthesis right square bracket
rightwards double arrow P R equals fraction numerator c cross times sin open parentheses alpha minus beta close parentheses cross times sin gamma plus c sin beta cross times sin open parentheses gamma minus alpha close parentheses over denominator sin open parentheses gamma minus alpha close parentheses end fraction
rightwards double arrow P R equals c open square brackets fraction numerator sin alpha cross times cos beta cross times sin gamma minus cos alpha cross times sin beta cross times sin gamma plus sin beta cross times sin gamma cross times cos alpha minus sin beta cross times sin alpha cross times cos gamma over denominator sin open parentheses gamma minus alpha close parentheses end fraction close square brackets
rightwards double arrow P R equals c open square brackets fraction numerator sin alpha cross times cos beta cross times sin gamma minus sin beta cross times sin alpha cross times cos gamma over denominator sin open parentheses gamma minus alpha close parentheses end fraction close square brackets
rightwards double arrow P R equals fraction numerator c sin alpha cross times open parentheses cos beta cross times sin gamma minus sin beta cross times cos gamma close parentheses over denominator sin open parentheses gamma minus alpha close parentheses end fraction
rightwards double arrow P R equals fraction numerator c sin alpha cross times sin open parentheses gamma minus beta close parentheses over denominator sin open parentheses gamma minus alpha close parentheses end fraction

Question 31

  

Solution 31

  

Chapter 10 - Sine and Cosine Formulae and Their Applications Exercise Ex. 10.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

C (a cos B – b cos A) = a2 – b2

Solution 5

Question 6

2(bc cos A + ca cos B +ab cosC)= a2 + b2 + c2

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

In a Δ ABC, prove that

sin3 A cos (B -C) + sin3B cos(C - A)+ sin3 C cos(A- B) = 3 sin A sin B sin C

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

I f space i n space a space capital delta A B C comma space cos squared A plus cos squared B plus cos squared C equals 1 comma space p r o v e space t h a t space t h e space t r i a n g l e space i s
r i g h t space a n g l e d.

Solution 15

C o n s i d e r space t h e space g i v e n space e q u a t i o n :
cos squared A plus cos squared B plus cos squared C equals 1
rightwards double arrow 1 minus sin squared A plus 1 minus sin squared B plus 1 minus sin squared C equals 1
rightwards double arrow 3 minus sin squared A plus 1 minus sin squared B plus 1 minus sin squared C equals 1

Question 16

Solution 16

Question 17

Solution 17

Question 18

b(c cos A – a cos C) = c2 –a2

Solution 18

Question 19

In any DABC, prove the following:

a cos A + b cos B + c cosC = 2b sin A sin C

Solution 19

  

Chapter 10 - Sine and Cosine Formulae and Their Applications Exercise Ex. 10VSAQ

Question 1

Solution 1

Question 2

 

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

I n space capital delta A B C comma space i f space a equals 8 comma space b equals 10 comma space c equals 12 space a n d space C equals lambda A comma space f i n d space t h e space v a l u e space o f space lambda.

Solution 6

B y space cos i n e space r u l e comma space w e space k n o w space t h a t comma
cos A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction ; space cos B equals fraction numerator a squared plus c squared minus b squared over denominator 2 a c end fraction ; space cos C equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction
T h u s comma space cos A equals fraction numerator 100 plus 144 minus 64 over denominator 2 cross times 10 cross times 12 end fraction equals 3 over 4
a n d
cos C equals fraction numerator 64 plus 100 minus 144 over denominator 2 cross times 8 cross times 10 end fraction equals 1 over 8
space cos A equals 3 over 4
rightwards double arrow A equals cos to the power of minus 1 end exponent open parentheses 3 over 4 close parentheses
rightwards double arrow A almost equal to 41.4096 degree.... left parenthesis 1 right parenthesis
cos C equals 1 over 8
rightwards double arrow C equals cos to the power of minus 1 end exponent open parentheses 1 over 8 close parentheses
rightwards double arrow C almost equal to 82.8192 degree..... left parenthesis 2 right parenthesis
F r o m space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space h a v e
C equals 2 A rightwards double arrow lambda equals 2

Question 7

Solution 7

Question 8

Solution 8

Question 9

In any triangle ABC, find the value of a sin (B -C) + b sin (C - A) + c sin (A - B)

Solution 9

Question 10

Solution 10

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