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Workshop  Calculation  &  Science  -  Electronics  Mechanic
                                              Exercise 1.6.26
                                       Exercise 1.6.26MechanicMechanic
 Workshop  Calculation  &  Science  -  Electronics  MechanicWorkshop  Calculation  &  Science  -  Electronics Workshop  Calculation  &  Science  -  Electronics
                                               Exercise 1.6.26
 Workshop  Calculation  &  Science  -  Electronics  Mechanic
 Trigonometry - Trigonometrical ratios
                                             Exercise 1.6.26
 Workshop  Calculation  &  Science  -  Electronics  Mechanic
 Trigonometry - Trigonometrical ratiosTrigonometry - Trigonometrical ratiosTrigonometry - Trigonometrical ratios
 Trigonometry - Trigonometrical ratios
 Trigonometry - Trigonometrical ratios
 Dependency
                                      1
                                1
                   sec
                      θ 
 Dependency
                                   
                             
 DependencyDependency
                                                 1 1
                                           1 AC
                               1
                                     AC
                                                          1
                  AC
                         1
                               AB
                          AB
           sec
                                        θ   θ 
 Dependency
                                     cos sec
                                               
                                 θ
               θ 
 The sides of a triangle bear constant ratios for a given
                                       1
                                 1
                          AC
 Dependency
                                                   θ   AB
                        AB
                             cos
                                     1 AB
                                                            θ
                                                        cos
                        AC
                               1 θ
                                          AB AB
                                                cos
                  AB θsec  
                              
                                   
 The sides of a triangle bear constant ratios for a givenThe sides of a triangle bear constant ratios for a givenThe sides of a triangle bear constant ratios for a given
                               AC
 definite value of the angle.  That is, increase or decrease in
                     θ 
                                 
                            
                 sec
                                AB
                          AB
                                     cos
                                         θ
 The sides of a triangle bear constant ratios for a given
                                          AC
                              AB
                                                   AC
                                       θ
                        AB
 definite value of the angle.  That is, increase or decrease indefinite value of the angle.  That is, increase or decrease indefinite value of the angle.  That is, increase or decrease in
 The sides of a triangle bear constant ratios for a given
 the length of the sides will not affect the ratio between them
                                AC
                                1
                         AB
 definite value of the angle.  That is, increase or decrease in
 the length of the sides will not affect the ratio between themthe length of the sides will not affect the ratio between themthe length of the sides will not affect the ratio between them
                              AC
 definite value of the angle.  That is, increase or decrease in
 unless the angle is changed. These ratios are trigonometrical
                                   
                   cot
                             
                      θ   
                                    AB
                        1
                                                          1
                                           1 AB
                  AB
                               1
                                                 1 1
 Workshop  Calculation  &  Science  -  Electronics  MechanicWorkshop  Calculation  &  Science
 unless the angle is changed. These ratios are trigonometricalunless the angle is changed. These ratios are tri -  Electronics
 the length of the sides will not affect the ratio between them  cotgonometricalunless the angle is changed. These ratios are trigonometrical Mechanic
                                    tanθ  θ cot Exercise 1.6.26Exercise 1.6.26
 the length of the sides will not affect the ratio between them
                               BC
                         BC
                                               
              θ 
                                           
                     
                                 θ   
                            cot
                          AB
 ratios.  For the given values of the angle  a  value of the ratios
                                1
                                      1
                       BC
                        AB
                                          BC BC
                                                        tanθ
                                    1 BC
                              1
                             tanθ
 unless the angle is changed. These ratios are trigonometrical
                  BC θ cot
                             
                                   
                        
                               AB
 ratios.  For the given values of the angle  a  value of the ratiosratios.  For the given values of the angle  a  value of the ratiosratios.  For the given values of the angle  a  value of the ratios
 unless the angle is changed. These ratios are trigonometrical
                                 
                    θ   
                           
                 cot
                               BC
                                     tanθ
                          BC
 Trigonometry - Trigonometrical ratiosTrigonometry - Trigonometrical ratios
 BC
 AC
 BC
 ratios.  For the given values of the angle  a  value of the ratios
 AB
                                                   AB
                        AB
 AB
                             BC
                                   tanθ
                        BC
 ratios.  For the given values of the angle  a  value of the ratios
 ,
 and
 ,
 ,
 ,
 AB
 ABAC
        AC
 ABBC
 AC
                         sideBC AB
 AC BCBC
 BC
 AC
 BC
 AB ABBC
 ACAB
                                 a
 , AC
 , AB
 , AB
 and ,,
    and
 ,
 and ACBC ,
 BC ,,
 ,
 ,,
                   sin
                              AB 
 AB
 BC
 AB
 AC
                     θ   
 AC
 BC
 do not change even whendo not change even whendo not change even when
                                            asideBC
                          a
                 sideBC
                                    sideBC
 AB
 BC AC
 BC ACAB
 ACAB
 AC BCAC
 AB BC and, AB
 AC BCAB
        BC
 AB
 BCAC
 AB ,
 AC BC,,
           sin
                         sideAC θ sin
 do not change even when
 DependencyDependency
                                         θ
                                          
                                 b 
                                                   
                            1
                     AC
                                  a 1 1
                                          1
                         sideBC AC
 the sides AB, BC, AC are increased to AB', BC' and AC' or
 and
 ,
 ,
 ,
 ,
 do not change even when
 BC
 AB
 AC
 AC
 AB
 BC
                          b
                                            bsideAC
                                    sideAC
                                                     b
              sec
                 sideAC θ sin sec
                                       
                       
                                a 
                       sideBC θ 
                  θ 
 AB
 AC
 the sides AB, BC, AC are increased to AB', BC' and AC' orthe sides AB, BC, AC are increased to AB', BC' and AC' orthe sides AB, BC, AC are increased to AB', BC' and AC' or
 AB
 BC
 BC
 AC
                 sin
 decreased to AB", BC" and AC".
                         side AB
                              
                    θ   
                                  c b θ  AB
                         sideAC
                                        cos
                              AB cosAB
                     AB
 The sides of a triangle bear constant ratios for a givenThe sides of a triangle bear constant ratios for a given
 the sides AB, BC, AC are increased to AB', BC' and AC' or
                       sideAC
                   cos
                                b
 decreased to AB", BC" and AC".decreased to AB", BC" and AC".decreased to AB", BC" and AC".
 the sides AB, BC, AC are increased to AB', BC' and AC' or
                           c
                      AB
                                    side
                  side
                                                 AB
                                             c side
                                                      c
                                   AC
                           AC
 For the anglealue of the angle.  That is, increase or decrease in
 definite value of the angle.  That is, increase or decrease indefinite v
                                          θ
                                             
           cos
                                      cos
 decreased to AB", BC" and AC".
               θ   
                                  b 
                                                    
                          sideAC θ cos
                              AB
                          side
                                   c
 decreased to AB", BC" and AC".
                        side b
                                             b sideAC
                                    sideAC
                  sideAC θ cos
 For the angleFor the angle
 For the angle
                            AB
                        
                                 c 
 the length of the sides will not affect the ratio between themthe length of the sides will not affect the ratio between them
                     θ   
                 cos
                               
 AC   is the hypotenuse
                                   b
                          sideAC
                                  1 1
                                          1
                           1 AB
                     AB
 For the angle
                          a
                        sideAC
                                 b
 For the angle
 unless the angle is changed. These ratios are trigonometricalunless the angle is changed. These ratios are trigonometrical
                      cot
                         θ  
                                      
              cot
                 θ   
                               
 AC   is the hypotenuseAC   is the hypotenuseAC   is the hypotenuse
                                     a
                   a
                                              a
 AB   is the adjacent side
                                 bBC
                                     a
                   sin
                          BC BC
                                        tanθ
                      θ   BC
                              a tanθ
 AC   is the hypotenuse
                          b
 ratios.  For the given values of the angle  a  value of the ratiosratios.  For the given values of the angle  a  value of the ratios
 AC   is the hypotenuse
                       a  a
                                   
                          b 
                                                         a
                                         a θ
                                                a a
                                       sin
            sin
 AB   is the adjacent sideAB   is the adjacent sideAB   is the adjacent side
               θ
                                            b
                              a x θ sin
                                              b
                                     b
                   b
                         a AB
                                   AB
                        x c
                                                   x
                                           x
                                     c a 
 BC   is the opposite side.
                  cos
                                 c b 
                                               
                      θ θ
 AB   is the adjacent side
                             b a
                    sin
                           b
 BC
 AB
 AB
 BC
 AC
 AB   is the adjacent side ACABABBCAC BC
 AC
                  sin
                             a
                                b
                     θ   c
                                            c c
                              c xθ
                                                         c
                                                c b
           cos
                         bc cos
                                         bθ
                       b 
 BC   is the opposite side.BC   is the opposite side.BC   is the opposite side.
                                                     c
                                    
                                    a c cos
               θ
 ,,
 and ,,
 ,
 ,,
 and
 ,
 do not change even whendo not change even when
                              x
                       
                                  
                            b c asideBC
                                     a
                    sideBC
 BC   is the opposite side.
                               b
                       θ
                                  c
 AB
 AB
 The ratios BCACAB AB
 BCAC
 AC BCAC
 BC
                                   
                 θ   
              sin
                      sin
                         c θ   
                                c
 BC   is the opposite side.
                             b
                     θ
                                              b
                                    c b
                 cos b
 The ratiosThe ratios
 The ratios
                             b sideAC
                    sideAC
                                     b
 the sides AB, BC, AC are increased to AB', BC' and AC' orthe sides AB, BC, AC are increased to AB', BC' and AC' or
                           b
 The ratios
                               BC
                         b side

 The ratios
                                        θ
                                    tan
                                  

                          =
 decreased to AB", BC" and AC".decreased to AB", BC" and AC".

                                          BC side

                                     side
                                                   BC
                        BC  AB side
                   side side
                                      c
                              c  AB
                             tan   AB
                            side
                                                        tan


                                               tan
                                                   θ

                                                           θ
                      cos
                                            =
              cos
                          θ   
                                             
                                                      
                            
                                θ =  

                               BC
                           side
                                                    AB
                             BC
                   side sideAC
                         side
                                      b
                               bsideAC
                         AB
                                    tan side
                                        θ   AB side

                                   
 For the angleFor the angle
                        =
                                   tan
                                 
                                      θ

                       =
                             1   AB
                           side
                              AB
                         side

                                                  or
                                 or
                                   cosec

                                                                 1
                                         θ 
                   sin

                      θ 
                                                              θ 
                                                       θ.cosec
                                                    sin

 AC   is the hypotenuseAC   is the hypotenuse AC  do not change even when    θ    =  θ   cos  θ    AC AC  sec   cos c 1 sin  AB AB θ     sin 1   θ   tanθ BC a b b    θ  Exercise 1.6.26Exercise 1.6.26
                                                         1
                                       11
                                                1
                                                                  1
                     1
                          or cosec   a
                      a
                                θ
                                            sincosec
                                  θ 
           sin
                                                               
                                               θ.cosec  θ cosec   or
                                      sin
                                                               sin

                                                          1 orθ
                                          θ
               θ 
                                                                                  θ 

                                                                                     1
                            cosec  θ  sin
                                                                         θ θ.cosec
 Workshop  Calculation  &  Science  -  Electronics  MechanicWorkshop  Calculation  &  Science  -  Electronics  Mechanic
                                          oror
                                                Exercise 1.6.26Exercise 1.6.26
                                                                            1
                                                                  θ.cosec  sin or

                                               1
                                     sin
                                                                  1 θ
                                         θθ  θ
                                                  or
                           1 b
                                                       θ.cosec
                                          a
                                    cosec
 AB   is the adjacent sideAB   is the adjacent side  sin sin     θ    θ  a θ  cosec    1 b  or a   or ba   cosec    1 cosec     θ sin   sin    θ  θ  θ  sin
                  cosec   θ  sin sin
                  θ
                      b
                                  cosec

                                        θ 
                                                     θ.cosec
                                                  sin
                                                                1

                                                or

                                θ
                                       
                                             sin
                           1 x
                                    x
                                                θ
                                         1
 Workshop  Calculation  &  Science
 Trigonometry - Trigonometrical ratiosTrigonometry - Trigonometrical ratios
                              θ

                                               θ
                        cosec
                              or
                                                       .   θ
                             c c
                                               cos

                                                     sec
                                sec

                                             or

                                    θ  cc
 BC   is the opposite side.BC   is the opposite side.  cos θ   cosc -  Electronics  sin Mechanic    θ  1     Exercise 1.6.26
                                  cb
                                                     WORKSHOP CALCULATION & SCIENCE - CITS
                      θ    b θ cos
                                                             1
                                               1
                                      1
                                                    1
                    1
                                  1
                         sec   θsec
                                          cos
                                                         θ
                                      orcos
                                              sec θ or
                       or

                                               .   θ θ sec
           cos

                                        cos orθ
              θ   
                             θ  θ cos 
                                          1
                           1
                          1 MechanicMechanic
 Workshop  Calculation  &  Science  -  Electronics Workshop  Calculation  &  Science  -
                                              or Exercise 1.6.26Exercise 1.6.26
                   cos Electronics
 The ratiosThe ratiosTrigonometry - Trigonometrical ratios b   or cos   sec   cos        or    sec        θ      .   θ  1    sec     θ or 1        θ cos       θ   .   θ    sec cos or    θ    1    . θ   sec     θ  1   
                  sec  b
                                                      cos
                                                            1    cos
                                                 θ
                                                       .   θ θ
                                        1 θ
                      θ θ  
                                 sec

                                    θθ 
                                                cos
                                                      sec

                               sec

                                                    sec
                             or
                 cos
                                              cos
                    θ   
                                   θ 
 DependencyDependency
                           1 θ
                          sec
                                         1 θ
                                                1
                           AC
                                        11
                                  1AC

                                      cos
          -  Electronics  Mechanic-  Electronics  Mechanic&  Science  -  Electronics  Mechanic
 Workshop  Calculation  &  Science Workshop  Calculation  &  Science Workshop  Calculation
                        sec
                      θ  θ  sec
                                                           θ
                                                      .   θ


                                                    tan
                                 cot
                                             or
                                                           1   
                                               cot
                   tan
 Trigonometry - Trigonometrical ratiosTrigonometry - Trigonometrical ratios sec   θ θ      tan tan or θ    θ  θ   θ        tanθ   θ           θ         .   θ θ cot 1   1 Exercise 1.6.26Exercise 1.6.26Exercise 1.6.26
                                                             1
                                                    1
                                      1
                                  1 BC

                    1 side
                           BC side
                                        cot 1 or θ  AB
                                       cos
                                 AB AB
                                             tan θ cos
                          cot θ   AB
                            cot
 Dependency
                                                        or
           tan
                                                           
                                                                              1   


                                                                 .   θ
 The sides of a triangle bear constant ratios for a givenThe sides of a triangle bear constant ratios for a given

                                                          cot
                                                                              θ
                                                               tanor
                                                          θ
                       or
               θ 
                                                    θ  or
                                                                          .
                                                                        tan
                                                                      1     θ
                                                                      θ
                                                                    cot
                                                   θ
                                                   1    cot
                    =
                                                    1
                            1 =
                                     or tan tanAC
                                tan cot   sec
                                  θ    MechanicMechanic
 Workshop  Calculation  &  Science  - Workshop  Calculation  &  Science

                       Electronics Electronics
                        side
                     θ -
                                        1 θ 
                                   θθ    cot 
           By pythogoras theorem we have,  AC² = AB² + BC² 2 1  θ  tan . θ  cot or  cot
                                                     .   θ Exercise 1.6.26Exercise 1.6.26
                                                  tan
                                                          tan
                                                θ
                                                              θ
                               or
                     θθ   
                  cot tan
 Trigonometry - Trigonometrical ratiosTrigonometry - Trigonometrical ratiosTrigonometry - Trigonometrical ratios cot      AB θ  AB        θ or   AB     cos    2 θ    θ  = AB  + BC 2
                          1   AB side AC
                                        AC
                 tan By pythogoras theorem we have,  AC
 definite value of the angle.  That is, increase or decrease indefinite value of the angle.  That is, increase or decrease in



                                                   tan
                             or
                                             cot
                                                         1   
                                                      θ
                             θ
                          cot
                                        tan

 The sides of a triangle bear constant ratios for a given
 DependencyDependency  By pythogoras theorem we have,  AC  = AB  + BCBy pythogoras theorem we have,  AC  = AB  + BCBy pythogoras theorem we have,  AC  = AB  + BC 2
                                                                               2
                                                          2
                                                                      2
                                              2
                                                                         2
                                                                            2
                                                    2
                                                                2

                                     1
                        cot AC

                                              1
                                         θ
                               1 AC
                                      tan 1
                           θ
 the length of the sides will not affect the ratio between themthe length of the sides will not affect the ratio between them
                                          1 AC
                             θ
                     θ 
                 sec
                        1 sec  
                   By pythogoras theorem we have,  AC  = AB  + BC
 definite value of the angle.  That is, increas
                                 1AB
                                        11
 The six ratios between the sides have precise definitions.e or decrease in AB    AB 1  θ cosec    AB    or θ  1 sin   1  2  2  2 θ θ.cosec     2  1  2 θ  1
                                                                 2
 Trigonometry - Trigonometrical ratiosTrigonometry - Trigonometrical ratios

                                                 

                                                  θ.cosec  sin or
                      sin
                  θ 
                             or
                               cosec  or
                          θ 
              sin
                 By pythogoras theorem we have,  AC  = AB  + BC
 unless the angle is changed. These ratios are trigonometricalunless the angle is changed. These ratios are trigonometrical AB
                              θ   cosAB
 DependencyDependencyDependency
                       θ    cot
                    cot
                         AC
                                            θ 1 1
                                      1 1 AC
                               1 AC
 The sides of a triangle bear constant ratios for a givenThe sides of a triangle bear constant ratios for a given
 The six ratios between the sides have precise definitions.The six ratios between the sides have precise definitions.The six ratios between the sides have precise definitions. θcosec      θ  tanθ  BC     cos θ    sin    θ 1
                           θ
                                        sin
                     cosec
 the length of the sides will not affect the ratio between them
                  sec
                           BC θ 
                                BC BC θsec
                     θ  sec
                                              tanθ 
                                  
 The six ratios between the sides have precise definitions.
 ratios.  For the given values of the angle  a  value of the ratiosratios.  For the given values of the angle  a  value of the ratios
                                      AC
                                       AB
 Opposite

 side
 BC
 definite value of the angle.  That is, increase or decrease indefinite value of the angle.  That is, increase or decrease in
                              AB
                                           cos
 The six ratios between the sides have precise definitions.  AB  AC cot AB θ    θ cos ABAB  1   θ   AB  1 cos θ
 unless the angle is changed. These ratios are trigonometrical
 SineThe sides of a triangle bear constant ratios for a givenThe sides of a triangle bear constant ratios for a given
 The sides of a triangle bear constant ratios for a given
 
 Sin
                                 AB
 BC θ  
 
 DependencyDependency
                                        AB
 θ   side Opposite
 OppositeBC
 side BC

 Opposite

          side
                                             1
                               1 AC
                       1
                                                  1
                                            BC
                                     1 BC
 the length of the sides will not affect the ratio between themthe length of the sides will not affect the ratio between them
     
 θ    BC
 Sine definite value of the angle.  That is, increase or decrease indefinite value of the angle.  That is, increase or decrease indefinite value of the angle.  That is, increase or decrease in sec   AC  sec   1AC   θ   or   11 1    AC   .   θ   sec cos or tanθ   θ    1     .  θ  sec     θ  1   
                          θ
 θ    AB
 Sin   
 AC
       Sin

 θ   side θ 
  BCBC ratios.  For the given values of the angle  a  value of the ratios θsec
                Sin
 ACSineABAB BCAC
                                           cos 
               cos
 AC
                                θ  sec   or θ 
          θ
                                      AC
 
                      cos
 Sine ACAB
 Hypotenuse
 BC
                   θ   θ   
                          or
 Opposite
                              1 AB
                                     1 1
                        AB
   ,
 ,
 ,
 ,
 BC ,,
 Opposite and,,
 and
                                       θ  θ cos AB
                                    AB
 θ
                             sec

 Sine
                                   cos
                                         AB
                     sec
 Sin
                 cot
 unless the angl  BC ACAB  AC BC side do not change even whendo not change even when   θ sideBC θ cot AB θ asideBC  a cos AB θ cos θ
 Hypotenuse AC
                                    
                     θ
 AC θ  e is changed. These ratios are trigonometricalunless the angle is changed. These ratios are trigonometrical
                               
 HypotenuseHypotenuseAC
 
 
 The sides of a triangle bear constant ratios for a givenThe sides of a triangle bear constant ratios for a given
 
 AB
 θ
 SinBC
 BCAC
 the length of the sides will not affect the ratio between themthe length of the sides will not affect the ratio between themthe length of the sides will not affect the ratio between them
 θ    ACAB AB
 Sine
                            sin BC
                                  tanθ BC
                               θ    BC
 Hypotenuse ABABBCAC
  AC
 AC BC
                               1 AB AC
                                             1 1
                                     1 1 ABAC
                         AB
 ratios.  For the given values of the angle  a  value of the ratiosratios.  For the given values of the angle  a  value of the ratios sin  θ    BC sideAC sideAC    b tanθ  1
 definite value of the angle.  That is, increase or decrease indefinite value of the angle.  That is, increase or decrease in
 AC AB
 ,
                                   b
 and
 Hypotenuse side Adjacent ,,,
 the sides AB, BC, AC are increased to AB', BC' and AC' orthe sides AB, BC, AC are increased to AB', BC' and AC' or θ cotlunless the angle is changed. These ratios are trigonometrica
 unless the angle is changed. These ratios are trigonometricalunless the angle is changed. These ratios are trigonometrica  1 cotl θ    1  θ cot 1 sideBC    1 a  
     do not change even when 
                              AB
                                      AB
 Cosine
 
 
 AB θ   AB
 CosBC
   AB
 AC
  θ   side Adjacent
 side AB BCAC
 Adjacent

 AdjacentAB
                                          θ   BC
                             cot BC
                                             tanθ
                                    
                                θ      or θ sin tanθBC
                                           cot
                                        or BC
                            BC
            side   tan
                                                      1     .  θ
                          or θ
                                     cot
                      tan
                                                  .   θ
                               
                                                       θ
 the length of the sides will not affect the ratio between themthe length of the sides will not affect the ratio between them
 BC
 θ    AC
 ratios.  For the given values of the angle  a  value of the ratiosratios.  For the given values of the angle  a  value of the ratiosratios.  For the given values of the angle  a  value of the ratios
 BCBC
 θ    θ    AC
  ABAC
  AB
 Cosine ACAB ABBC
 θ   Cosine
                      θ
                  Cos
 Cosine decreased to AB", BC" and AC".decreased to AB", BC" and AC".   Cos θ      θ   cot    θ side AB θ sidesideAC c    BC tancot or 1   tanθ  tan       θ  1   
 Hypotenuse
 AC
  
 AB
                                              θ b
 Adjacent
                                   tan c 1AB
 Cos side
                                      θ  AB 11
                             cot  AB

                                           tan


 ,
 ,,
 ,
 and
 ,
 AB the sides AB,  and,,,
 side
  do not change even whendo not change even when
                                               
                                              AB
                                  
                               θ   θ
                    cossideBC AB
                                           
 unless the angle is changed. These ratios are trigonometricalunless the angle is changed. These ratios are trigonometrical
 AdjacentBC, AC are increased to AB', BC' and AC' or θ cot
                        θ    cos asideBC AB  cot
 AC θ  
 
 
 Cosine
 HypotenuseHypotenuseAC
                                        a
 Hypotenuse AC
  AC BCAC
 Cosine ACAB AB
 θ   BCAB
 AB For the angleFor the angle Hypotenuse ACAB ABBC   BC Cos θ    AC  By pythogoras theorem we have,  AC  = AB  + BCBy pythogoras theorem we have,  AC  = AB  + BC 2
 BCAC
 θ
 Cos
                                           b BC
                                   BC
                                    b  tanθBC
                    θ   
                           sideAC sideAC
                          sin
                                                tanθ
                                BC
                             θ
                 sin
 AB
 AC BC
 AC AB ABBC ACBC
                                                       2
 AB
 AC
                                                               2
 BC
                                                             2
                                                         2
 BC AC
                                                 2
 decreased to AB", BC" and AC".
 ratios.  For the given values of the angle  a  value of the ratiosratios.  For the given values of the angle  a  value of the ratios
                                                c
 ,
 ,
 , ,
 ,
                                        b
 and
 , ,
 ACBC and,,
 , ,
 and ,
 Hypotenuseside Opposite do not change even whendo not change even whendo not change even when
                                cos
 the sides AB, BC, AC are increased to AB', BC' and AC' orthe sides AB, BC, AC are increased to AB', BC' and AC' or  sideAC  bsideAC  side  AB   a
                        sideBC
                                   θ    sideBC
                                a sideBC
                                        a AB
                                   AB
 AB
 
   BCAC
 
 Opposite BCAC ABAB
 side BC BCAC
 BC θ   ACBCAC ABAB
 Tangent
 AB
 Tan BC
 BCAC
                                  sin
             side sin

 OppositeBC
 AC   is the hypotenuseAC   is the hypotenuse
                             θ
                          sin
   θ   side Opposite
                                    θ
                                
                                       sideAC 
                                                b
                                        
                     θ   
                                                      2
 For the angle
 The six ratios between the sides have precise definitions.The six ratios between the sides have precise definitions.  Dividing both sides of the equation by AC , we have  2  2
 AB
 AC
 AC
 AB
 BC
 BC
 decreased to AB", BC" and AC".decreased to AB", BC" and AC". ACABABBCACBC
                                   a
                            a
         
   
 θ   Tangent
  side
           Tan
 Adjacent
 θ    θ   
              θ
 AB
 
 θ   
 Tangent
 Tangent
                                         c
                                 c b side
                            AB sideAC
                                              2 b
                                     AB bsideAC
 BC
 Tan side
                       θ   side
 Opposite
                   Tan sideAC
           Dividing both sides of the equation by AC , we haveDividing both sides of the equation by AC , we haveDividing both sides of the equation by AC , we have
 and
 ,
 , ,
 ,
 , ,
 ,
 and,
 the sides AB, BC, AC are increased to AB', BC' and AC' orthe sides AB, BC, AC are increased to AB', BC' and AC' orthe sides AB, BC, AC are increased to AB', BC' and AC' or
 side do not change even whendo not change even when
 BC
                          cos
                     θ   
                                
                 cos
                                        
                                             a
                             θ  sideBC sideBC
                                      a
                      2 θ
                            sin
                     sin
                                              a
 AB   is the adjacent sideAB   is the adjacent side
 Tan
 Adjacent 
                                a θ
 AB θ
                                  2 b
                                       aa
 Tangent
                                          b
 
                            2th sides of the equation by AC
           Dividing both sides of the equation by AC², we have , we have
 AC   is the hypotenuse
   AC
 BC
 AB AB
                                                      2
 AB
 AC AB
 side AB BCAC BC
 side
                       sin
                                      a b 
                                   b 
 
                                 b sideAC
 Tan
  θ
                        sideAC sin
 θ   OppositeOppositeBC
 BC
 
 Tangent

                   Dividing bo b 
 side
                          θ
                                 θ   
 For the angleFor the angle Opposite        AC AdjacentAB BC  θ   side Adjacent   side Dividing both sides of the equation by AC , we have
                   AC
                              
                                            
 decreased to AB", BC" and AC".decreased to AB", BC" and AC".decreased to AB", BC" and AC".
                                                    2
 AB
 side
 Adjacent
                                                 c
                                         x c  AB
                         side
                                        side
                               BC x  c  ABside
                         AB   AB
                           2 sideAC sideAC
 Sine θ    Sine θ    Adjacent    Sin    Cosec   θ  AC  2  cos 2 θ    2 θ  =  BC cθ cos  +   2 cos  b cb  2 BCAB cca 2 b  b 2   a  2
 side θ
 Sin
   θ
 AB AC 
 Hypotenuse
                                       2
 the sides AB, BC, AC are increased to AB', BC' and AC' orthe sides AB, BC, AC are increased to AB', BC' and AC' or
                                   c c
                                                   BC
                                  2 AB AC θ  
 BC   is the opposite side.BC   is the opposite side.
                  AB cos
                                  2 θ
 θ    AB   is the adjacent side
                            2 AC b θ cos sin
 Cosecant
 
 AC
 HypotenuseHypotenuseAC
 Hypotenuse
 HypotenuseHypotenuseAC
                         AC
 AC   is the hypotenuseAC   is the hypotenuse
                                        b bsideAC
                               ACbsideAC
 AC
 AC
                            2
                      2
 For the angleFor the angleFor the angle
                   AC
                          2 2 AB
                                         2 
                                       2 
                                 2 a
                 AC Cosec  a
 Cosecant decreased to AB", BC" and AC".decreased to AB", BC" and AC".    Cosec  = AC  + sideAC + AC 2 BC = b ACAC  ABside + =  2 x  2  + b AC  2
 Opposite
 Cosec  θ
                        AB θ
 side
 BC
 Cosecant Cosecant
              2
   
                                          ACAC c
                 θ
                                       c
 θ   
                    2 2
 θ 
 
                              BC  ABside
 θ   
    
 AC
 Hypotenuse
                        AC b
                  AC
                        =
           AC
                                        c
 BC   is the opposite side.
                            +  cos
 ACos
                                cos θ
 The ratiosThe rati
 Hypotenuse
                            2
 θ
                      2
 BC

 side BC
 
 Opposite 
 Cosec  side  Opposite
 Cosecant
                                            a
                                AC ab
                          sin
                  sin
                                        b
 AB   is the adjacent sideAB   is the adjacent side   side side   OppositeBC  θ  θ    side  AC 2 AC θ   =cos θ   2 2 AC a θ  AC x a 2 θ    a  b x a  b  b c  c
 AC   is the hypotenuseAC   is the hypotenuseAC   is the hypotenuse
                         b
                                 2 2 b
 Cosecant
 Cosec
                              sideAC sideAC
 For the angleFor the angle
 
 θ   
 
 AdjacentAB
 side
                        a
 AB
  Adjacent
 Opposite
 BC
                                   
                        AC 
                  2 ⎡
                          2 ⎡
 Hypotenuse
 AC
                                    2
                                        b
                                             22
 BC
 Cos
 
 
 θ    The ratios
 θ
 Cosine θ   Cosine θ      Opposite    side HypotenuseHypotenuseAC  θ Cos θ    ⎡ AB⎤ sin ⎡ θ   AB⎤ c cosθ sin b a θ  BC⎤ BC side    ⎡ AB cc b    aa  b BC⎤ a 2
                           + side c

                                        BC
 BC   is the opposite side.BC   is the opposite side.
 AB   is the adjacent sideAB   is the adjacent sideAB   is the adjacent side
 
                                c
                  = θ
 Secant
                  cos
                                     b a θ
                                AB c ab sin
 Sec
 AC
 Hypotenuse
 AC
                      BC⎤ 2 b
                    ⎢
                                 =
                        ⎥ =
                                         BC⎤ ⎤ tan
                                 ⎥ 2⎤ tan
  AC
 HypotenuseHypotenuseAC
 
                    + AC 
 side
 Adjacent
                             =
 θ θ 
 AB Secant
 Secant  θ    AC   is the hypotenuseAC   is the hypotenuse      Sec  ⎢   ⎥ Sec  ⎡ ⎢ θ  2 ⎦ ⎤   ⎢ ⎡ side ⎡ b   +=x  ⎡ θ  b   + x    ⎡ θ   ⎥
 Sec  
           = θ
 θ   Secant
                    ⎣ AB
                              AC x  a
                                2 ⎦ ⎤ side a
 Hypotenuse
 AC
                            ⎣ BC   AB
                                                ⎢
                                       ⎢ ⎢   AB
                                   ⎥
 AC
 Hypotenuse
                                        side
                         ⎥ b c
                                        ⎣ c c
 Secant
 side AB
                                     b θ
 AB θ
                                                 AC⎦ c
             ⎣
              AC⎦ cos
 Sec
 AdjacentAB
                                         AC⎦ ⎦ cb
 BC   is the opposite side.BC   is the opposite side.BC   is the opposite side.
                                                ⎣ c
                            BC⎤ AC⎦ ccb θ
 
 Adjacent 
 The ratiosThe ratios AB   Adjacent    side  side side  Sec  θ  θ side  Adjacent    side  =  ⎡ = θ  ⎣ sin + ⎥ θ   ⎡ cos 2⎢ b ⎢ b c cos 2=  b ⎣ AC    ⎥ ⎥ BC  a tan    θ
                    AB⎤ AC⎦ +
                                      b
                                             b
                                 ⎥ aθ
                               ⎣ sin
 AB   is the adjacent sideAB   is the adjacent side
                    ⎢
                                         aa
 
 θ   
 Secant
                               ⎥ ⎦
                       ⎥ ⎦
                   ⎢ 1 =  (cos θ ⎣
                    ⎣
 AB
                      AC
                              AC
                                             AB
 Adjacent  OppositeBC

 Adjacent
                           ⎢)  + (sin θ) x
                                        
   side
 BC
                                    
 AB Opposite
                                      1 side x
                           ⎣
                             2 1
                                                        1
                                                1
 side
                   ⎣
                    AC⎦ b
                            AC⎦ b
                                       2 b
  Cot
 θ
 Cotangent
 
                                      BC  c
 TangentosThe rati   θ    θ  e rat     side AB Tan  AdjacentABTan  θ   θ  sideAdjacent Dividing both sides of the equation by AC , we haveDividing both sides of the equation by AC or   sin   θ.cosec   θ  1
 
 Adjacentios
 θ   TangentosTh
                                                                  

                     2
                                                  θ

 The rati BC   is the opposite side.BC   is the opposite side.
                                   or b θ

                        θ  sin BC  side θ
                               θ c cos
 AB
                                                 c
                                           θ cosec    c
                                          c
                                     cosec  or b

               side sin cos
           1 =  (cos θ)  + (sin θ)1 =  (cos θ)  + (sin θ)1 =  (cos θ)  + (sin θ) , we have 1θθ.cosec   sin or  side c
                                                22 2
                                                        2 2
                                   tan
                                 
                                               θ
                     Cot =
                  sin θ  +  cos  θ   = 1
 Opposite
                           2  + (sin θ)
 BC
                               =
   θ   
                   1 =  (cos θ) cosec
                                 θ
                    
                         θ
 Cotangent    θ     AB  Cotangent Cotangent side      Cot  θ    1 =  (cos θ)  + (sin θ)   ABside 2 θ    θ   tan    sin    θ  sin    θ  1
                              2
                                   2 cosec
 Cot side  
 θ     θ   
                     2
                             2
  Adjacent
 side
 AdjacentAB
    side
 AB
 Adjacent
                                         2 2  1
                               AB
                         side
 AB
 Adjacent
 Cotangent
  Cot

           sin θ  +  cos  θ   = 1 sin θ  +  cos  θ   = 1sin θ  +  cos  θ   = 1 θcosec   orθ  sin side b
 side

                              BC
                                      2 BC side
 
                      2
 The ratiosThe ratios
 
 Cotangent BC     θ       θ Opposite   BC   Opposite   BC side OppositeBC  θ    θ   sideOpposite 2  side    2  2  AB AC =   side    b 2 2    BC θ  tan =     tan       BC  2  tan    θ    or   sin   θ.cosec   θ  1
 Cot
                              2
                        2
                          2
                           BCAB =
              AC
                                               θ 
 side
                                           1 θ

                                                             θ
                                                          sin

                   2  sin θ  +  cos  θ   = 1 1 cosec
                              2
                                                   1
                            21
 Relationship between the rat     side  sin θ  +  cos  θ   = 1,  Tangent,  Cosec, Sec  and
 Oppositeios
                                         side
                         + = 1   ABside
 AC
                     Sine,  Cosine + 1   ABside
                                                 θ 1   AB
 BC HypotenuseHypotenuseAC
                                                          1
                   =
                              2 θ
                                      2 θ  sec   or

                                                       sec cos
                                                 cos
                                                        .   θ or
 Relationship between the ratiosRelationship between the ratiosRelationship between the ratios AC sin    cos 2 θ    cos  2 sec   or   BC   AC θ  θ cosec   or BC         θ  or tan   sin θ   θ.cosec  sin or    θ   1    sec  .  θ θ θ.cosec     1    θ  1     θ  1
                             θ  side
                                  cosec   side
 Cosec  
 
                 2
                          sin
                                or

                                                orθ
 
 θ  
         θ
                          2
  θ Cosec
 Cosecant Cosecant
 θ   
                     θ 
              Sine,  Cosine,  Tangent,  Cosec, Sec  andSine,  Cosine,  Tangent,  Cosec, Sec  andSine,  Cosine,  Tangent,  Cosec, Sec  and

                                             θ
                     ACAC sec
                                         cos
                                                 cos
                           ACAC θ
                                  sec
                     Cotangent are the six trigonometrical ratios
                                                    θ
                                        tan

                        cosec   =
                                    =
 1een the ratios  BC
                              θ cosec

                                        1 sin
 1 side Opposite

                                       θ
                                               θ
 Relationship betw OppositeBC
                                                    sin
                                                       1 θ
 AC
                      Sine,  Cosine,  Tangent,  Cosec, Sec  and
                                            or 1
                                cos 1
                                                  θ  1
                                                              1
                            1
                                               1
                                   θ    side
                                              sec
                                                             cos
                                                                    .   θ
                                                           or
                                                                         1   
                                                                   sec

                                    AB
 Relationship between the ratios  side  1  Cotangent are the six trigonometrical ratiosCotangent are the six trigonometrical ratiosCotangent are the six trigonometrical ratios
                               side
                                                                         θ
                                            AB

 Cosec
 
 θ 
 
                    Sine,  Cosine,  Tangent,  Cosec,  Sec  and
                                                                         1
                     θ  sin
 AC
 1
                                orsin
                                  cosec  orθ   
                  sin


                                                 orcosec
                                                   sin
                                                                             θ 
 1 1
                                                                 1
                                         θ cosec
                                                                     θ
                                                      θ.cosec   sin orθ 


                                                θ or
 1 AC
                                                              θ θ.cosec   θ.cosec   sin or

 1
 AC
                                                                                1
                             θ 
                     2
                                                         θ
                                           1 θ

                                                  1 cos
                                     2sec
 BC
                      Cotangent are the six trigonometrical ratios
                                    1
                             2 2 1
 sin
 BC
 θ    θ
 θ
 Cosec  θ     AC  Cosec  Cosec         Sine,  Cosine,  Tangent,  Cosec, Sec and  Cotangent  are the six trigonometrical  ratios
                                                               θ

                                               θ
 1
                                       θ
                                                            sin
                             Sin  cosec   cosec

                                                    2    sin
                                           2  sinθ
                                                       θ
                               θ
 1
                         cosec
                ⎡
                                 BC⎤
                                                1
                                ⎡ θ  1
                        ⎡AB 1
                                    and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1   cot or  θ cot  or 1
                         BC⎤
                 AB⎤
 AC HypotenuseHypotenuseAC

                                                                    θ
                        ⎡ 
                                                                    1   
                    Cotangent are the six trigonometrical ratios 1  θ  tan . θ  cot or  θ cot  orθ  tan
                                       
                                                            tan . θ
                     sin θ = θ =
                         θ =
 AC
                                   θ   or 1
                      Sin  cos  1
 1 θ
 1
 BC θ 
                             orθ
                                                                   θ
                                        sec
                                                            sec
                                                    or
                                        Sin  cos or  θ 
                 cos
                                                      cos

                       + = 

 Sec
 θ 
 Secant θ  BC   θ  sin AC   BC  BCBC sin θ  BC  θ sin θ    tan θ = θ =  ⎥ θ   tan θ = θ =  ⎤   tan θ + sec     ⎢  cot  + cos ² θ = θ =   Sin  sec  . θ      1 θ or   θ    1    . θ 1 θ.cosec   1     θ  1 2   
 
  Sec
              =
 
 Cosec  θ  Secant
  

                                              = 1 tan
                ⎢

 Cosec
                            and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1 and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1 and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1 1θθ.cosec   sin orθcosec   orθ  sin

                                    2  θ   cosec   or
                           θ
                       sin
                                                      1 sin
                                         tan
                           cot


                            ⎥ ⎥ 
                                            2  θ θ 
                    =
                                tan θ = θ =
                                                      2
                 θ =
                                                               2 2
                           and sin ² θ = θ =  1 θ  cosθ
                                                  θ
                        ⎢ ⎢ sec
                            Cos   θ =  tan θ = θ =    cosθsec
                              Sin 
 side
 AB
  Adjacent
 AdjacentAB
                 AC⎦
                        ⎣
                ⎣
 AC BC  BC BC  sin sin θ    AC side   AC  tan θ = θ =  By pythogoras theorem we have,  AC  = AB  + BCBy pythogoras theorem we have,  AC  = AB  + BC1  θsec  . θ  cos or  sec   orθ cos ⎥   Cos   θ =  Cos     cos  1     .  θ    θ cot    θ    .  tan  1     2
                         AC⎦ ⎦
                                         1 1
                        ⎣AC 1
                                                 1 sin
                            Sin  ⎣ tan 1 θcosec   cosec
                                          2  θθ θθ θ + cos  θ θ  θ θ  θ = 1 or  θ cot  or θ
 BC
 θ
                                                    2  θ
                                                   
                                                        1   sin
                                    and sin
                                                             2 2  θ
                                 AC⎦ θ   
                                                                   2 2  θ   
                      tan θ  θ =
                     Cos θ = θ =
                                            2
                     Cos  = θ =

                                sec
                                                       2
                  cos
                          cos

                                               cos

                             θ
                                              θ
                                                            sec
                                                           θ
                                                  2  sec  . θ or  θ 
                                                                  1   
                              or
                                            or
                     θ   
                                   θ  sec   or
                       θ = =
                       θ  θ =
                                  and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1 θ  tanθ  cot
 AC AC              tan θ = θ =  Cos   + (sin θ) 1 θ  cosθ Cos    2 sec   θ  cos  cos    θ
                        sec
                                sec
                                2 2 1 θ
                                                1 θ
                        2 1
 side
 AB  AdjacentAB   Adjacent    side  1 =  (cos θ)  + (sin θ)1 =  (cos θ)  θ    or 1    1  tan or 1  61
                                  1
                                                         1    . θ
                                                          θ
                                                        cot
                                By pythogoras theorem we have,  AC  = AB  + BC


   θ   
 CotangentCotangent    θ        Cot  θ      Cot  θ    tan    θ   tan    orθ  cot θ    or   sec   cot   θ   or cot       θ or   .   θ      cos         .   θ or   tan  θ   1   sec  .    θ  1     2    θ  1     2  61  2  61
 The six ratios between the sides have precise definitions.The six ratios between the sides have precise definitions. cos  cos
                                         θ  sec   or
                                                          61 θsec cos
                          θ
                         2 θ
                         2  cot
                                 cot

 side
              sin  +  cos² θ  +  cos  θ   = 1 11
                                 θ
 BC  OppositeBC    Opposite     side  sin² θ  +  cos  θ   = 1sin  = 1 sec   2    θ  tan    θ θ 1 cos tan 1 θ    θ  cos     1 θ  61
                                  1 sec
                 2
              2
                      2
                          tan
                                        cot

                                                    2    θ  cot 
                                   θ      orθ   
 The six ratios between the sides have precise definitions.
                 By pythogoras theorem we have,  AC  = AB  + BCBy pythogoras theorem we have,  AC  = AB  + BC
 BC  OppositeOppositeBC    side    side  sin θ +  cos θ =1 orθ  tan    cot    θ    θ       cottan θ  tancot   θ    oror cotcot          θ    .   θ    θ    tanor  tan   θ     θ 2 1   tan .  or    2   cot   61 2 1   θ   . θ    tan       θ 2 1 
                                                             Cosine of an angle= Sine of its complementary
                                                                2
                                          θ θ
                                              tan

                                 cot

 θ    Sine
 
 Sine
 
 θ   
 Sin
           It can be transformed as 1
 Relationship between the ratiosRelationship between the ratios  θ      Sin θ   It can be transformed as  1  1  1  angle
 AC
   side
 Opposite

                  By pythogoras theorem we have,  AC  = AB  + BCBy pythogoras theorem we have,  AC  = AB  + BCBy pythogoras theorem we have,  AC  = AB  + BC
                                                         tancot
 HypotenuseHypotenuseAC BC
                                                     2
                 Sine,  Cosine,  Tangent,  Cosec, Sec  andSine,  Cosine,  Tangent,  Cosec, Sec  and 2
 The six ratios between the sides have precise definitions.The six ratios between the sides have precise definitions.  tan    θ  tan    θ or    cot  θ      or  cot     θ    or    cot         .   θ or 2    2   θ    1    tan .  θ  2  2       θ  1     2  2  2
                                               θ
                                     cot
                                                       θ




                                            tan
                              cot
                                 θ
                                                   tan
                                         θ
 AC  1 AC Sine θ     1     Sin θ    Cotangent are the six trigonometrical ratiosCotangent are the six trigonometrical ratios
 1 1
                    sin θ = 1 – cos θ
                      2
                                   2
 Hypotenuse
                                                           Examples
  
 Cosec  θ Cosec    AB  AdjacentAdjacentAB AC    side side     side  By pythogoras theorem we have,  AC  = AB  + BCBy pythogoras theorem we have,  AC  = AB  + BC 2
 The six ratios between the sides have precise definitions.The six ratios between the sides have precise definitions.The six ratios between the sides have precise definitions.
  sideOpposite
 OppositeBC

 θ BC
                                                                     2 2
                                                          2
                                                                2 2
 θ
 Sine BC  Sine  sin θ   BC  sin  θ   Sin   θ    Cos Sin  θ    θ   Cos θ    Sin   Sin   2   2  2   2
 θ
 θ   Cosine
 BC BC
 θ   
 Cosine
 
                                and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1 and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1
                 tan θ = θ =  tan θ = θ =
                             θ = =
                    θ = =
                    θ  θ =
                    sin θ =
 BC

 sideBC
 AC AC HypotenuseHypotenuseAC
       side
 OppositeOppositeBC
 side
 AC
 Adjacent
   Opposite
                               –
                              1
 AC HypotenuseHypotenuseAC AB
 The six ratios between the sides have precise definitions.The six ratios between the sides have precise definitions.  θ  θ =   cos 2   0   1  0
      side
 Cosine θ Sine
 Sine θ    Sine θ    θ       Sin θ   Sin θ     Sin  θ   θ    Cos    Cos    If sin 30 =  2   find the value of sin 60
 
            Cos
 Hypotenuse
 HypotenuseHypotenuseAC
 AC  BC AdjacentAB  AC   side Hypotenuse  or cos θ = 1 – sin θ
 AC

 AB BC
  side
 OppositeOppositeBC
                      2
                                  2
 OppositeOppositeBC side Adjacent
 side side

 Tan  θ
 Cosine Sine θ    θ    θ    Sine θ   θ            Cos   θ  Sin side  Tan θ θ    Dividing both sides of the equation by AC , we haveDividing both sides of the equation by AC , we have
                                                                     61
     θ    θ Cos  Sin
 θ   Cosine
                                                           By applying pythagores theorem
 TangentTangent
                                                             61 2
                                                       2
 AC AC
 AB
 sideAB
 AdjacentAB

 side Opposite side Adjacent

 AdjacentAdjacentAB BC
                                   2
 TangentCosine
  θ
 θ
 θ
 Cosine θ   Cosine AB HypotenuseHypotenuseAC HypotenuseHypotenuseAC  Adjacent   side side side   θ     Cos Tan θ   θ   cos θ =  2 2Dividing both sides of the equation by AC , we have
     Cos
 θ    
    
                                    
                               –
                                 sin
  Cos
 
                              1
                                                           BC  = AC - AB
                                                                        2
                                                              2
                                                                   2  2
                                    2 2
 AC  AC OppositeBC AdjacentAB  HypotenuseHypotenuseACside Opposite Hypotenuse  AC  2  AB AC  BCAB  BC 2
 HypotenuseHypotenuseAC
 AC
 AB

       side
 Adjacent
   Adjacent
 AB
 side side
 side
 BC

                                       +
 CosecantCosecant   
 θ
 Tangent Cosine θ    θ   Cosine θ     Tan Cosec    Tan θ Cos  θ Cosec θ Cos  θ  θ  Dividin 2  =  ACAC  + =  2 2 2    -   1 AB 2 2 2  BC 2  2  2
  θ  
 θ   Tangent
 θ   
                             2 2
                    ACg both sides of the equation by AC , we haveDividing both
                                ACAC sides of the equation by AC , we have
                                       cos AC
                             sin AC
 AdjacentAB HypotenuseHypotenuseAC
   Opposite
 BC

   side
 AB  BC AC  Opposite AC side  Hypotenuse side Opposite  tan θ =    =    =  +
 OppositeBC
 side
  side
 BC
 sideAdjacent BC
   side Opposite
                                
                             cos
                                               2
  Tan
             
 Cosecant Tangent
               Tan
 Tangent  θ   Tangent θ   θ    θ      θ     Tan   θ    Cosec   θ θ    2  2th sides of the equationividing both sides oD 2 2 2 2  cos  2  2 AC by AC , we havef the equation by AC , we haveth sides of the equation by AC , we have
                  Dividing boD
                                ACividing bo
                                                     2
                                                             2
                                                                     2
                           2 2
                                 2
                                       AC 2
       side

                        AB AC
                              BCAB
 Opposite
 BC
  Adjacent

 side
 AB  BC HypotenuseHypotenuseAC    sideAB   side    side side  AC  ⎡ = AB⎤  + = BC⎤ ⎤  + ⎡ BC
 AC AdjacentAdjacentAB
                              ⎡ AB
                             ⎡
                                       BC⎤
 HypotenuseHypotenuseAC OppositeOppositeBC
 AC
 θ  
 θ   Secant
 Sec
                                   +
 Secant
                           =
 θ      Tangent
 θ
 Tangent
                                 2
 Cosecant Cosecant  θ   θ          Cosec   θ Tan θ   Sec  θ  θ θ   AC = 2 2 Dividing bothDivid ⎥ ⎥ sides of the equation by AC , we haveing both sides of the equation by AC , we have
    θ Cosec  Tan
                           2
                                sin 2
                          ⎥
                                         2
                            + 2
                                          ⎥
                              ⎢ ⎢ACAC
                                                          2
                                                                  2
                                      ⎢ 2 AC
                      ⎢ ACAC
                                               2 2
                                         2 2
                           2 2
                                 2 2 2
                              BCAB AC
                        AB AC
   Adjacent
 AdjacentAB
 side

 AB
                       AC
                      ⎣
                                          ⎡
                                       AC⎦ BC⎤
                             ⎣
                              ⎣ AC
                                 ⎡

 sideOpposite

 BC  AB  AC side  Hypotenuse  AC tan θ = ⎦  + = AC⎦ ⎦ AB⎤ ⎣ BCAB  +  BC
 OppositeBC AdjacentAdjacentAB
 ACside
 side side
                                    2+ =
                       =
 HypotenuseHypotenuseAC
 AC
  Hypotenuse
                              1
 θ   
 Secant
                                     
                     θ
             θ Cosec
 Cosecant Cosecant θ  Cosecant θ        Adjacent    side side     Sec  θ  AC 1 =  (cos θ)  + (sin θ)1 =  (cos θ)  + (sin θ)⎦ 2  2
                                        + 2 2
                                 ⎢
                               – = 2
 
                     2
                                 sin 2 2
 Cosec  
 θ   
     θ Cosec  
                           2 2
                                          ⎢
                                      2 ⎥ACAC
                                              2 ⎥ AC
                        2 ACAC
                              ACACAC
                                         2
 AB
 Adjacent
                                   AC⎦
     side

                                     2 2 2
                                           AC
                              2 22 2 2
                          2
                                          ⎣ BC
                              AB AC
                       AC
                                 ⎣ BCAB
 AdjacentABside OppositeBC
 BC
 AB OppositeBC
 side side Opposite
                   ⎡
                                    ⎡
                    AB⎤
                           ⎡
 HypotenuseHypotenuseAC
                           ⎡ AB
 AC
 CotangentCotangent AC
   θ   HypotenuseHypotenuseAC
               θ
         
           Cot
   θ   
    
 
 Secant CosecantCosecant  θ       Sec  θ   Cot  θ    We know sin θ + cos θ = 1 += 2 2 = BC⎤ ⎤ 2 2 + = BC⎤  +  2  2  BC  = 2 - 1 2
                          +
  θ  
         θ
 θ   Secant
     Sec
                    2
                                                              2
                                                                 2
                           ⎢
    Cosec  
                                      2 2
          θ Cosec
                 = θ
 θ   
                              2  ⎥ 2
                   ⎢
                       2  ⎥ 2 ⎢ ACAC
                               2 ⎥ 2 ⎢ ACAC
                                       2 ⎥ 22 AC
 BC
    Opposite
 side
                    sin  + cos² θ  +  cos  θ   = 1
           We know sin² θ  +  cos  θ   = 1sin  = 1
                                          2
 AB  AdjacentAB  OppositeBC  AC  Opposite    side   side  ⎣ AB AC  ⎣ BC⎦ 1 =  (cos θ)  + (sin θ) 2
 AB

 sideAdjacent
                            AC
 side
 Adjacent
                                    ⎣BCAB
                                    ⎡ ⎡
                            ⎡ ⎣ AB
                                           ⎡
                                             BC⎤
                    AC⎦ ⎤
                   ⎡
                            AC⎦ ⎤ ⎤
                                     AC⎦ ⎤ ⎤
                           ⎡
 HypotenuseHypotenuseAC
 Hypotenuse

      side
 side
 BC
 AC
 OppositeBC
               Cot
 Cotangent
                   θ
 
              
 Secant  θ   Secant θ  Secant θ     θ       Sec  θ   Sec  Dividing both sides by cos θ. += +  2  ⎥ ⎥  = 2  ⎢ ⎢  ⎥ ⎥  +               = 4 - 1
         θ 
                  = θ
             Sec
                   ⎢
                       ⎥

 Opposite
 BC
                                sin . 22
                           2⎢ ⎢ 2
         side
                                           2⎢ 2
                                     AC⎦
 AB
 sideAB
                           ⎣
                                    ⎣ ⎣
                    AC⎦
                                                ⎦
                 1 =  (cos θ)  + (sin θ)1 =  (cos θ)  + (sin θ) ACAC⎦
                   ⎣
 AdjacentAB
 side
                                           ⎣
                             AC⎦ ⎦
                            ⎣ AC
                                   22
                                            2
 side
 Adjacent

 Relationship between the ratiosRelationship between the ratios side AdjacentAB  Adjacent   Adjacent    side  Dividing both sides by cos² θ  +  cos  θ   = 1 ⎥
 AB
                       Sine,  Cosine,  Tangent,  Cosec, Sec  andSine,  Cosine,  Tangent,  Cosec, Sec  and
                                ⎡
                          AB⎤
                        ⎡
                                          BC⎤
                                        ⎡
                                  BC⎤ ⎤
 HypotenuseHypotenuseAC
 AC
   θ   
 Cotangent Cotangent   θ        Opposite Cot side θ  Cot  θ    2 θ  2 = ⎢  2 2  ⎥  = 1 ⎢ ⎢ ⎡ AB 22 ⎥ ⎥  + ⎢  2 2 ⎥         = 3
       
 
    θ
 θ   Secant
          Sec
  Sec
                               +
 Secant
 
 
 θ  
                      θ Cotangent are the six trigonometrical ratiosCotangent are the six trigonometrical ratios

 side
 BC
 OppositeBC
                  cos
                  1 =  (cos θ)  + (sin θ)  θ   = 1  + (sin θ)
               θ sin²θ  +  cos  θ   = 1sin
                                    2
                            2
 11
                          1 =  (cos θ) 1 ACAC⎦
  Adjacent
   Adjacent
 ABAB
                                   Sine,  Cosine,  Tangent,  Cosec, Sec  and
 1AC AdjacentAB
 AC Relationship between the ratios  AdjacentAB
 AdjacentAB
                                 ⎣ AC
   side
                                        ⎣
    side
 1 side
                                     ⎦ ⎦
                                ⎣
                                          AC⎦
           side
                         =
                    +
 Cosec  θ Cosec   CotangentCotangent  θ      side   Cot  θ      Cot sin  2   sin 2  θ    2 +  ⎣ Cos²θ  +  cos  + (sin θ)1 =  (cos θ)  2
  
 θ
                               =
 Cotangent
             θ
   θ
   θ   
                 Cot
                                2
                             cos
 θ   BC
                                  Cotangent are the six trigonometrical ratios
                                                           BC =
 sin
 sin
           cos
 θ
                      θ
 sideBC

                                      Sin 
   side
                                 Cos²θ  +  cos
                  sin
                         Cos²θ  +  cos  θ   = 1sin
 BC BC
 BC BC Opposite OppositeBC
                                             2   θ   = 1
                                     2
 AC
 1
                  cos 2
 1
                                             2
 Relationship between the ratiosRelationship between the ratios    Opposite     side  θ  Cos²θ  +  cos  θ   = 1sin 2  2  Sin  2   and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1 and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1 and
                       1 =  (cos θ)  + (sin θ)1 =  (cos θ)  + (sin θ)
                                         2
                                                 2
                                 2
                                        2
 AB
 side
 AdjacentAB
   Adjacent
        side

                          θ  θ =
                          θ = =
                                                             2
                    Sine,  Cosine, Cosine,  Tangent,  Cosec, Sec
                       tan θ = θ =  tan θ = θ =  Tangent,  Cosec, Sec  andSine,
                                                     2
                                                    2
                                  θ = =
                                  θ  θ =
 θ 
 Cosec
 AC  θ   
 CotangentCotangent    AC  θ    BC      Cot  θ      Cot  θ    2   Cos   Cos  
   θ
 sin
                               2
                    Cotangent are the six trigonometrical ratiosCotangent are the six trigonometrical ratios
                                           Sin 
 Relationship betwRelationship between the ratiosRelat BC BC OppositeBC     Opposite     side  or 1 + tan θ = sec θ 22   tan θ = θ =   Cosine,  Tangent,  Cosec, Sec  and
 side
                 or 1 + tan² θ  +  cos  θ   = 1sin
                       sin  = sec² θ  +  cos  θ   = 1
 1een the ratios 11ACionship between the ratios
 1
                         2
                                          2
 AC
                                                                0  θ θ  θ θ  θ = 1
                                      θ = =
                                                 and sin  θθ θθ θ + cos
                                      θ  θ =
                                                         2
                    Sine,  Cosine,  Tangent,  Cosec, Sec  andSine,  Cosine,  Tangent,  Cosec, Sec  andSine,
                                                                 2
 AC
 θ
 Cosec  θ Cosec       θ   BC    sin θ    Using the same equation  Cos    sin 60  =
 BCBC
 sin
                    Cotangent are the six trigonometrical ratiosCotangent are the six trigonometrical ratiosCotangent are the six trigonometrical ratios
 BC
                                                                          61
                                                                  61
                                    Sin 
                            Sin 
 1 1 AC
 1 AC
 AC
 Relationship between the ratiosRelationship between the ratios
                                θ  θ =
                       θ  θ =
                        θ = =  Sine,  Co
                                           and sin  θθ θθ
                                   and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1θ + cos  θ θ  θ θ  θ = 1
                                θ = =
                                                  2  2
                                                           2
                                          2
 Cosec  θ Cosec  Cosec      1 1    1  2   tan θ = θ =  tan θ = θ = sine,  Tangent,  Cosec, Sec  andSine,  Cosine,  Tangent,  Cosec, Sec  and
 θ
  θ
 AC
 AC
                      2
                           Cos  
 sin
 BC  AC  sin θ   BC BC 11  θ   BC  1 sin θ    sin θ + cos θ = 1.  Cos    2   Sin   2  2   2  2   3 2   61
 BC BC
                          Cotangent are the six trigonometrical ratiosCotangent are the six trigonometrical ratios
                                    Sin 
                            Sin 
 1AC
                                                           Cosθ 
                                θ = =
                                        θ  θ =
                                θ  θ =
                        θ  θ =
                        θ = =
                                        θ = =
                                                                     Find the other trigonometrical ratios
                    tan θ = θ =  tan θ = θ =  tan θ = θ =
 Cosec  θ Cosec   AC    AC  Dividing both sides by sin θ,   and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1 and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1 and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1
 AC θ 
 
                                                                  5
                              2
                                    Cos  
                                            Cos  
                            Cos  
 BC  BC BC  sin θ   BC  sin θ     Sin   Sin                   61      61
                             θ  θ =
                                     θ  θ =
                                     θ = =
                             θ = =
                                                        2
                                                2
                                                                2
                                                           83
                                                       2
                     2
 AC  AC          cos     tan θ = θ =  tan θ = θ =   and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1 and sin  θθ θθ θ + cos  θ θ  θ θ  θ = 1
                                                           By applying pythagores theorem
                               1
                                 Cos   Cos  
                 1      =                                      61      61      61
                    2
                  sin       sin 2         CITS : WCS - Electrical - Exercise 7
                                                           AB  = AC  - BC
                                                                   2
                                                                        2
                                                             2
                                                                     61      61
                1 + cot θ = cosec θ                                      =  5 - 3      =  25 - 9
                                                                      2
                                                                  2
                      2
                                 2
                1 + tan θ = sec θ                                 = 16
                      2
                               2
      Trigonometrical Tables
           Ratio   0 o   30 o  45 o  60 o   90 o
                          1     1      3
            sin θ   0                        1
                          2      2    2
                                                           AB    =  16  = 4
                          3     1     1
           cos θ    1                        0
                          2      2    2                    Now
                          1
            tan θ   0           1      3    ∞
                          3
       When θθ θθ θ increases,
       Sine value increases;
       Cosine value decreases;
       Tangent value increases to more than 1 when the
       angle is more than 45  (tan60  = 1.732)
                                   o
                            o
        Sine of an angle = Cosine of its complemen-
                            tary angle
                                                                            st
      62         WCS - Electronics Mechanic : (NSQF - Revised Syllabus 2022) - 1  Year : Exercise 1.6.26
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