RD SHARMA Solutions for Class 11-science Maths Chapter 5 - Trigonometric Functions

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Chapter 5 - Trigonometric Functions Exercise Ex. 5.1

Solution 1
Solution 2

Solution 3
Solution 4
Solution 5
Solution 6
Solution 7
Solution 8
Solution 9

Solution 10

Solution 11

Solution 12

 
Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

fraction numerator 2 space sin space theta over denominator 1 plus cos space theta plus sin space theta end fraction equals x
rightwards double arrow fraction numerator 2 space sin space theta left parenthesis 1 minus cos space theta plus sin space theta right parenthesis over denominator left parenthesis 1 plus cos space theta plus sin space theta right parenthesis left parenthesis 1 minus cos space theta plus sin space theta right parenthesis end fraction equals x space space space left square bracket R a t i o n a l i z i n g space t h e space d e n o m i n a t o r right square bracket
rightwards double arrow fraction numerator 2 space sin space theta left parenthesis 1 minus cos space theta plus sin space theta right parenthesis over denominator left parenthesis 1 plus sin space theta right parenthesis squared minus cos squared space theta end fraction equals x
rightwards double arrow fraction numerator 2 space sin space theta minus 2 space sin space theta cos space theta plus 2 space sin squared theta over denominator 1 plus sin squared space theta plus 2 sin space theta minus cos squared space theta end fraction equals x
rightwards double arrow fraction numerator 2 space sin space theta left parenthesis 1 plus cos space theta minus sin space theta right parenthesis over denominator 2 sin squared space theta plus 2 space sin space theta end fraction equals x
rightwards double arrow fraction numerator 2 space sin space theta left parenthesis 1 plus cos space theta minus sin space theta right parenthesis over denominator 2 sin space theta left parenthesis 1 plus sin space theta right parenthesis end fraction equals x
rightwards double arrow fraction numerator 1 plus cos space theta space minus sin space theta over denominator 1 plus sin space theta end fraction equals x space left square bracket C a n c e l l i n g space t h e space 2 space sin space theta space i n space b o t h space N u m e r a t o r space a n d space D e n o m i n a t o r right square bracket
H e n c e space P r o v e d

Solution 18

Solution 19

square root of fraction numerator a plus b over denominator a minus b end fraction end root plus square root of fraction numerator a minus b over denominator a plus b end fraction end root
equals square root of fraction numerator begin display style a over b end style plus 1 over denominator begin display style a over b end style minus 1 end fraction end root plus square root of fraction numerator begin display style a over b end style minus 1 over denominator begin display style a over b end style plus 1 end fraction end root space left square bracket D i v i d i n g space b o t h space N u m e r a t o r space a n d space d e n o m i n a t o r space b y space b right square bracket
equals square root of fraction numerator tan space theta plus 1 over denominator tan space theta minus 1 end fraction end root plus square root of fraction numerator tan space theta minus 1 over denominator tan space theta plus 1 end fraction end root
equals square root of fraction numerator begin display style fraction numerator sin space theta over denominator cos space theta end fraction end style plus 1 over denominator begin display style fraction numerator sin space theta over denominator cos space theta end fraction end style minus 1 end fraction end root plus square root of fraction numerator begin display style fraction numerator sin space theta over denominator cos space theta end fraction end style minus 1 over denominator begin display style fraction numerator sin space theta over denominator cos space theta end fraction end style plus 1 end fraction end root
equals square root of fraction numerator begin display style fraction numerator sin space theta plus cos space theta over denominator cos space theta end fraction end style over denominator begin display style fraction numerator sin space theta minus cos space theta over denominator cos space theta end fraction end style end fraction end root plus square root of fraction numerator begin display style fraction numerator sin space theta minus cos space theta over denominator cos space theta end fraction end style over denominator begin display style fraction numerator sin space theta plus cos space theta over denominator cos space theta end fraction end style end fraction end root
equals square root of fraction numerator sin space theta plus cos space theta over denominator sin space theta minus cos space theta end fraction end root plus square root of fraction numerator sin space theta minus cos space theta over denominator sin space theta plus cos space theta end fraction end root
equals fraction numerator sin space theta plus cos space theta plus sin space theta minus cos space theta over denominator square root of sin squared space theta minus cos squared space theta end root end fraction
equals fraction numerator 2 space sin space theta space over denominator square root of sin squared space theta space minus space cos squared space theta end root end fraction

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

L H S space equals space a b plus a minus b plus 1
equals left parenthesis s e c space theta minus tan space theta right parenthesis left parenthesis cos e c space plus c o t space right parenthesis plus s e c space theta minus tan space theta minus cos e c space theta minus c o t space theta plus 1
equals open parentheses fraction numerator 1 over denominator cos space theta end fraction minus fraction numerator sin space theta over denominator cos space theta end fraction close parentheses open parentheses fraction numerator 1 over denominator sin space theta end fraction plus fraction numerator cos space theta over denominator sin space theta end fraction close parentheses plus fraction numerator 1 over denominator cos space theta end fraction minus fraction numerator sin space theta over denominator cos space theta end fraction minus fraction numerator 1 over denominator sin space theta end fraction minus fraction numerator cos space theta over denominator sin space theta end fraction plus 1
equals fraction numerator 1 over denominator sin space theta space cos space theta end fraction plus fraction numerator 1 over denominator cos space theta end fraction cross times fraction numerator cos space theta over denominator sin space theta end fraction minus fraction numerator sin space theta over denominator cos space theta end fraction cross times fraction numerator 1 over denominator sin space theta end fraction minus tan space theta cross times c o t space theta plus fraction numerator 1 over denominator cos space theta end fraction minus fraction numerator sin space theta over denominator cos space theta end fraction minus fraction numerator 1 over denominator sin space theta end fraction minus fraction numerator cos space theta over denominator sin space theta end fraction plus 1
equals fraction numerator 1 over denominator sin space theta space cos space theta end fraction plus fraction numerator 1 over denominator sin space theta end fraction minus fraction numerator 1 over denominator cos space theta end fraction minus 1 plus fraction numerator 1 over denominator cos space theta end fraction minus fraction numerator sin space theta over denominator cos space theta end fraction minus fraction numerator 1 over denominator sin space theta end fraction minus fraction numerator cos space theta over denominator sin space theta end fraction plus 1
equals fraction numerator 1 over denominator sin space theta space cos space theta end fraction minus fraction numerator sin space theta over denominator cos space theta end fraction minus fraction numerator cos space theta over denominator sin space theta end fraction
equals fraction numerator 1 over denominator sin space theta space cos space theta end fraction minus fraction numerator sin space theta over denominator cos space theta end fraction minus fraction numerator cos space theta over denominator sin space theta end fraction
equals fraction numerator 1 minus sin squared space theta minus cos squared space theta over denominator sin space theta. cos space theta end fraction
equals fraction numerator 1 minus left parenthesis cos squared space theta plus sin squared space theta right parenthesis over denominator sin space theta. space cos space theta end fraction
equals fraction numerator 1 minus 1 over denominator sin space theta. cos space theta end fraction equals 0 equals R H S. space H e n c e space P r o v e d

Solution 25

Solution 26(i)

Solution 26(ii)

Solution 26(iii)

Chapter 5 - Trigonometric Functions Exercise Ex. 5.2

Solution 1(i)

Solution 1(ii)
Solution 1(iii)

Solution 1(iv)
Solution 2
Solution 3

Solution 4
Solution 5

Chapter 5 - Trigonometric Functions Exercise Ex. 5.3

Solution 1(i)
Solution 1(ii)
Solution 1(iii)
Solution 1(iv)
Solution 1(v)
Solution 1(vi)
Solution 1(vii)
Solution 1(viii)
Solution 1(ix)
Solution 1(x)
Solution 1(xi)
Solution 1(xii)
Solution 1(xiii)
Solution 1(xiv)
Solution 2(i)
Solution 2(ii)

Solution 2(iii)

Solution 2(iv)
Solution 2(v)
Solution 2(vi)

Solution 2(vii)
Solution 3(i)
Solution 3(ii)
Solution 3(iii)
Solution 3(iv)
Solution 3(v)
Solution 4

Solution 5
Solution 6(i)
Solution 6(ii)
Solution 6(iii)
Solution 7
Solution 8(i)
Solution 8(ii)
Solution 9(i)
Solution 9(ii)
Solution 9(iii)
Solution 9(iv)
Solution 9(v)

Chapter 5 - Trigonometric Functions Exercise Ex. 5VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution 5

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution 6

  

 

 

 

 

 

 

 

 

 

 

 

 

Solution 7

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution 8

The maximum value for sin(x) is 1 for all x.

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution 9

  

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution 10

The radius of circular wire is 15 cm.

The circumference of circular wire is .

The radius of loop is 120 cm.

The circumference of loop is .

The circumference of a circle covers all the quadrants of the circle i.e. the angle subtended by it at the centre is .

The circular wire is cut and bent so as to lie along the circumstance of the loop.

So, circumference of circular wire covers  

  of the circumference of the loop.

Therefore the angle subtended by the circular wire is

 .

 

Solution 11

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution 12

  

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution 13

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution 14

Solution 15

  

 

 

 

 

 

So the least value of  is 2.

Solution 16

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution 17