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Workshop  Calculation  &  Science  -  Electronics  Mechanic
                                                                                               Exercise 1.6.25
              Workshop  Calculation  &  Science  -  Electronics  Mechanic
             Workshop  Calculation  &  Science  -  Electronics  Mechanic                      Exercise 1.6.25                                Exercise 1.6.25
                                                            Trigonometry - Measurement of angles
              Trigonometry - Measurement of angles
             Trigonometry - Measurement of angles
                                                     WORKSHOP CALCULATION & SCIENCE - CITS
                                                            Introduction:
              Introduction:
 Workshop  Calculation  &  Science  -  Electronics  Mechanic                 Exercise 1.6.25                          Circumference makes an angle (2πr) = 360°
                                                                       Circumference makes an angle (2πr) = 360°
             Introduction:
                                                                       Circumference makes an angle (2πr) = 360°
                                                            Trigonometry is the branch of mathematics which deals
                                                                                                                      Radius of the circle makes an angle (r) = 1 Radian
              Trigonometry is the branch of mathematics which deals
                                                                              Exercise 1.6.25
 Workshop  Calculation  &  Science  -  Electronics  Mechanic
                                                                       Radius of the circle makes an angle (r) = 1 Radian
           EXERCISE 7: Trigonometrydeals
 Trigonometry - Measurement of angles                       with the study of measurement and relationship of the                 C       360 o
             Trigonometry is the branch of mathematics which
                                                                       Radius of the circle makes an angle (r) = 1 Radian
              with the study of measurement and relationship of the
             with the study of measurement and relationship of the
                                                            three sides and three angles of a triangle. o
                                                                                    C
 Trigonometry - Measurement of angles
              three sides and three angles of a triangle.
                                                                                            360 o
 Workshop  Calculation  &  Science  -  Electronics  Mechanic            Exercise 1.6.25                                               ie :     r  =   1Radian
                                                                                           360
                                                                                    C
             three sides and three angles of a triangle.
                                                                                        =
                                                            Units:                     ie :
 Introduction:
                                                     Circumference makes an angle (2πr) = 360°
                                                                                          1Radian
                                                                                       =
                                                                                       ie :
                                                                                    r
              Units:
            Trigonometry - Measurement of angles
                                                                                          1Radian
                                                                                    r
             Units:
 Trigonometry - Measurement of angles                Radius of the circle makes an angle (r) = 1 Radian                          2π r     360 o
 Trigonometry is the branch of mathematics which deals
 Introduction:
                                                            Measurement of Angles
                                                      Circumference makes an angle (2πr) = 360°

              Measurement of Angles
 with the study of measurement and relationship of the                       o    2π r   r    360  o                                  =
                                                                                  2π
             Measurement of Angles
                                                                                           360 o
 Trigonometry is the branch of mathematics which deals
                                                            There are three systems of measuring the angle:
                                                      Radius of the circle makes an angle (r) = 1 Radian
                                                                                        =
 three sides and three angles of a triangle.
              There are three systems of measuring the angle:
 Introduction:  Introduction:                                        ie :    C  =    360  r  =   1Radian                          r     1Radian
 with the study of measurement and relationship of the     Circumference makes an angle (2πr) = 360°
             There are three systems of measuring the angle:
                                                            (i) Sexagesimal System r
                                                                              o
                                                                        1Radian
                                                                           360
 three sides and three angles of a triangle.
                                                                  rC
 Units:
           Trigonometry is the branch of mathematics which deals with the study of measurement and relationship of the
              (i) Sexagesimal System
 Trigonometry is the branch of mathematics which deals      Radius of the circle makes an angle (r) = 1 Radian 1Radian                    360 o
             (i) Sexagesimal System
                                                                                               o
           three sides and three angles of a triangle.
                                                            This is called British System.  In this system, one right
 with the study of measurement and relationship of the                      ie :     r  =   1Radian  360                          2π =
                                                                                            360 o
 Measurement of Angles
 Units:
                                                                             o
              This is called British System.  In this system, one right
                                                                         360
                                                                        o
                                                                2π
                                                                   r
                                                                                    2π =
                                                                     360
             This is called British System.  In this system, one right
                                                             C
 three sides and three angles of a triangle.                angle is divided into 90 equal parts which are called                        1Radian
                                                                                          1Radian
                                                                                   2π =
           Units:
              angle is divided into 90 equal parts which are called
                                                                      =
                                                                                          1Radian
 There are three systems of measuring the angle:                      ie :    degrees.Each part is divided into 60 parts which are called
                                                                =
 Measurement of Angles
                                                                              o
             angle is divided into 90 equal parts which are called
                                                                        1Radian
                                                                  r r  1Radian
              degrees.Each part is divided into 60 parts which are called r
 Units:    Measurement of Angles                                  2π      360                                              2π  Radian =  360°
                                                                            2π  Radian =  360°
             degrees.Each part is divided into 60 parts which are called
                                                                       =
                                                                            2π  Radian =  360°
 There are three systems
 (i) Sexagesimal Systemof measuring the angle:              minutes.Each minute is divided into 60 parts which are
              minutes.Each minute is divided into 60 parts which are
                                                            called seconds.The parts so divided respectively are called:
 Measurement of Angles minutes.Each minute is divided into 60 parts which are  r 360 o  1Radi oan                           π Radian = 180°
                                                                          360
           There are three systems of measuring the angle:
              called seconds.The parts so divided respectively are called: r 2π
                                                                              π Radian = 180°
 (i) Sexagesimal System
 This is called British System.  In this system, one right       2π =        π Radian = 180°
             called seconds.The parts so divided respectively are called:
                                                                =
                                                                              o
 There are three systems of measuring the angle:            One degree (1°), one minute (1') and one second (1")             1 Radian =   180 o
                                                                        1Radian
                                                                           360
           i  Sexagesimal System
                                                             r
                                                                  1Radian
                                                                                             o
 angle is divided into 90 equal parts which are called
              One degree (1°), one minute (1') and one second (1")
 This is called British System.  In this system, one right
                                                                                          180 o
                                                                   2π =
                                                            It means 1 right angle 1 Radian =   180
 degrees.Each part is divided into 60 parts which are called
 (i) Sexagesimal System One degree (1°), one minute (1') and one second (1") 2π  Radian =  360°adian = 90° (90 degrees)                   π
                                                                         1R
                                                                              1 Radian =
 angle is divided into 90 equal parts which are called
           This is called British System.  In this system, one right angle is divided into 90 equal parts which are called
                                   = 90° (90 degrees)
              It means 1 right angle
                                                                        o
 minutes.Each minute is divided into 60 parts which are              360                   π π
             It means 1 right angle
                                  = 90° (90 degrees)
                                                                           1 degree (1°) = 60' (60 minutes)
 degrees.Each part is divided into 60 parts which are called
           degrees.Each part is divided into 60 parts which are called minutes.Each minute is divided into 60 parts which
 This is called British System.  In this system, one right  2π  Radian =  360°             π                                             π
                             1 degree (1°) = 60' (60 minutes)
                                                            π Radian = 180°
                                                            2π =
 called seconds.The parts so divided respectively are called:
           are called seconds.The parts so divided respectively are called: 1Radian
                                                                                          π
 minutes.Each minute is divided into 60 parts which are
                            1 degree (1°) = 60' (60 minutes)
 angle is divided into 90 equal parts which are called                     1 minute (1') = 60" (60 seconds)                       1° =  180 o   Radian
                                                                                    1° =
                                                                                            o   Radian
                                                                                         180 o   Radian
                             1 minute (1') = 60" (60 seconds)
                                                             π Radian = 180°
 One degree (1°), one minute (1') and one second (1")
 called seconds.The parts so divided respectively are called:
 degrees.Each part is divided into 60 parts which are called  2π  Radian =  360°  180 o  1° =  180                Examples
                            1 minute (1') = 60" (60 seconds)
           One degree (1°), one minute (1’) and one second (1”)
                                                            1 Radian =
                                                                   Examples
 minutes.Each minute is divided into 60 parts which are     In Trigonometry, mostly this system is used.
              In Trigonometry, mostly this system is used.
                                                                   Examples
                                                                            o
                 = 90° (90 degrees)
 It means 1 right angle
 One degree (1°), one minute (1') and one second (1")
                                                                         π 180
             In Trigonometry, mostly this system is used.
           It means 1 right angle
                                                            (ii)  Centesimal System
 called seconds.The parts so divided respectively are called: = 90° (90 degrees) π Radian = 180°                  1 Convert 45°36’20” into degree and decimal of degree.
                                                             1 Radian =
                                                                   1 Convert 45°36’20” into degree and decimal of degree.
                                                                         π
              (ii)  Centesimal System
                1 degree (1°) = 60' (60 minutes)                   1 Convert 45°36’20” into degree and decimal of degree.
                  = 90° (90 degrees)
 It means 1 right angle
                                                                           π
             (ii)  Centesimal System
                          1 degree (1°)
                                                                      o
                                                                          o   Radian
                                                                  1° =
 One degree (1°), one minute (1') and one second (1")  = 60’ (60 minutes)  This is called French System.  In this system, the right  60 second = 1 minute
                                                                   180
                                                                             60 second = 1 minute
                1 minute (1') = 60" (60 seconds)
                1 degree (1°) = 60' (60 minutes)
              This is called French System.  In this system, the right
                                                                       180 π 60 second = 1 minute
                                                            angle is divided into 100 equal parts which are called
             This is called French System.  In this system, the right
            = 90° (90 degrees)
 It means 1 right angle                 1 minute (1’)   = 60” (60 seconds) 1 Radian =   1° =  180 o   Radian  20           20 second =   20   = 0.333’
                                                 Examples
                                                                    π
              angle is divided into 100 equal parts which are called
 In Trigonometry, mostly this system is used.
                1 minute (1') = 60" (60 seconds)
                                                            grades.  Each grade is divided into 100 minutes and each
             angle is divided into 100 equal parts which are called
                                                                             20 second =  20
                                                                                              = 0.333’
              grades.  Each grade is divided into 100 minutes and each
                1 degree (1°) = 60' (60 minutes)  1 Convert 45°36’20” into degree and decimal of degree.                                60
                                                                    π
           In Trigonometry, mostly this system is used.
                                                                            20 second =
                                                                                              = 0.333’
                                                  Examples
             grades.  Each grade is divided into 100 minutes and each
                                                            minute is divided into 100 seconds. 60
 (ii)  Centesimal System
 In Trigonometry, mostly this system is used.
                                                                     o   Radian
                                                             1° =
              minute is divided into 100 seconds.
                                                                                         60
                1 minute (1') = 60" (60 seconds)  1 Convert 45°36’20” into degree and decimal of degree.                    60 minute = 1 degree
                                                                  180
             minute is divided into 100 seconds.
           ii   Centesimal System
                                                                             60 minute = 1 degree
                                                           60 second = 1 minute
                                                                             60 minute = 1 degree
 This is called French System.  In this system, the right Examples  Parts so divided are respectively called:
 (ii)  Centesimal System
              Parts so divided are respectively called:
 In Trigonometry, mostly this system is used.               60 second = 1 minute          36.333                        36.333 minute =   36.333   = 0.606°
             Parts so divided are respectively called:
           This is called French System.  In this system, the right angle is divided into 100 equal parts which are called
 angle is divided into 100 equal parts which are called
                                                            One grade (1 g), one minute (1' ), one second (1").
                                                                        20
                                                                                          36.333
 This is called French System.  In this system, the right
              One grade (1 g), one minute (1' ), one second (1"). 20 second =
                                                                            = 0.333’
                                            1 Convert 45°36’20” into degree and decimal of degree.
                                                                          36.333 minute =
           grades.  Each grade is divided into 100 minutes and each minute is divided into 100 seconds.
                                                                                                  = 0.606°
 grades.  Each grade is divided into 100 minutes and each
 (ii)  Centesimal System One grade (1 g), one minute (1' ), one second (1").  It means 1 right angle = 100 grades (100g)                  60
                                                                         36.333 minute =
                                                                                            60   = 0.606°
 angle is divided into 100 equal parts which are called
                                                                        60 20
              It means 1 right angle = 100 grades (100g)
 minute is divided into 100 seconds.                  60 second = 1 minute    = 0.333’     60
                                                            20 second =
           Parts so divided are respectively called:
             It means 1 right angle = 100 grades (100g)
 grades.  Each grade is divided into 100 minutes and each
                                                                                                                              0
 This is called French System.  In this system, the right  60 minute = 1 degree 45 36’20” = 45.606°                         45 36’20” = 45.606°
                                                                           1 grade (1 g) = 100 minutes (100’)
                                                                         60
                                                                                0
                             1 grade (1 g) = 100 minutes (100’)
 minute is divided into 100 seconds.
 Parts so divided are respectively called:
                                                                             45 36’20” = 45.606°
 angle is divided into 100 equal parts which are called                    1 minute (1') = 100 seconds (100")
                                                                   20
                                                                                0
           One grade (1 g), one minute (1’ ), one second (1”).
                            1 grade (1 g) = 100 minutes (100’)
                                                            60 minute = 1 degree
                                                                        36.333
 grades.  Each grade is divided into 100 minutes and each  20 second =  60   = 0.333’    = 0.606°                 2 Convert 24.59° into degree, minute and second
                                                                   2 Convert 24.59° into degree, minute and second
                             1 minute (1') = 100 seconds (100")
 Parts so divided are respectively called:
 One grade (1 g), one minute (1' ), one second (1").
                            1 minute (1') = 100 seconds (100")
                                                                   2
           It means 1 right angle
                                                                                        90° = 100g (because each is a right
 minute is divided into 100 seconds.   = 100 grades (100g) 36.333 minute =  Convert 24.59° into degree, minute and second    1 degree =   60 minute
                                                                          60
                                                                         36.333
                                          90° = 100g (because each is a right
                                                      60 minute = 1 degree
 It means 1 right angle = 100 grades (100g)  = 100 minutes (100’) 36.333 minute =   1 degree =   60 minute
 One grade (1 g), one minute (1' ), one second (1").
                                                                                                angle)
                                         90° = 100g (because each is a right
                                                                                 = 0.606° =   60 minute
                          1 grade (1 g)
 Parts so divided are respectively called:        angle)    45 36’20” = 45.606° 1 degree                                  0.59 degree =   0.59 x 60   = 35.4’
                                                                           60
                                                              0
                1 grade (1 g) = 100 minutes (100’)  angle)  This system is easier than Sexagesimal System.  But to
                                                                            0.59 degree =   0.59 x 60   = 35.4’
 It means 1 right angle = 100 grades (100g)
                                                                   36.333
                                                                           0.59 degree =   0.59 x 60   = 35.4’
                          1 minute (1’)
 One grade (1 g), one minute (1' ), one second (1").  = 100 seconds (100”)  use this system many other systems will have to be devised  1 minute =   60 second
                                                  36.333 minute =
                                                             45 36’20” = 45.606°
                                                                           = 0.606°
              This system is easier than Sexagesimal System.  But to 0
                                                 2 Convert 24.59° into degree, minute and second
                1 minute (1') = 100 seconds (100")
             This system is easier than Sexagesimal System.  But to
                1 grade (1 g) = 100 minutes (100’)
                                                                              1 minute =   60 second
                                                                     60
              use this system many other systems will have to be devised
                                       90°
 It means 1 right angle = 100 grades (100g)  = 100g (because each is a right angle)  1 minute =   60 second                0.4 minute =   60 sec  x   0.4
                                                            that is why this system is not used.
             use this system many other systems will have to be devised
                             90° = 100g (because each is a right
                                                  2 Convert 24.59° into degree, minute and second
                                                            1 degree =   60 minute
                1 minute (1') = 100 seconds (100")
              that is why this system is not used.
                                                                             0.4 minute =   60 sec  x   0.4
                                                         0
             that is why this system is not used.
           This system is easier than Sexagesimal System.  But to use this system many other systems will have to be
                1 grade (1 g) = 100 minutes (100’)  angle)  45 36’20” = 45.606°  0.4 minute =   60 sec  x   0.4                       =  24”
                                                            (iii)   Circular System
                             90° = 100g (because each is a right
                                                          0.59 degree =   0.59 x 60   = 35.4’
              (iii)   Circular System
                                                             1 degree =   60 minute
                                                                                        =  24”
           devised that is why this system is not used.
             (iii)   Circular System
                1 minute (1') = 100 seconds (100")  angle)  2 Convert 24.59° into degree, minute and second=  24”     Therefore 24.59° =  24 35’24”
 This system is easier than Sexagesimal System.  But to
                                                            In this system, the unit of measuring angles is radian.  It
                                                                                                                                          0
                                                            1 minute =   60 second
                                                           0.59 degree =   0.59 x 60   = 35.4’
              In this system, the unit of measuring angles is radian.  It
                                                                        Therefore 24.59° =  24 35’24”
           iii  Circular System
 use this system many other systems will have to be devised
                                                            is that angle which is formed at the centre and is formed
                             90° = 100g (because each is a right  1 degree =   60 minute   0 0
             In this system, the unit of measuring angles is radian.  It
                                                                       Therefore 24.59° =  24 35’24”
 This system is easier than Sexagesimal System.  But to
              is that angle which is formed at the centre and is formed
                                                            of an arc of length equal to radius in a circle.
                                                           0.4 minute =   60 sec  x   0.4
 that is why this system is not used. angle)formed at the centre and is formed 1 minute =   60 second             3 Change 50 37’30” into degrees
             is that angle which is
           In this system, the unit of measuring angles is radian.  It is that angle which is formed at the centre and is formed
                                                                                                                              0
 use this system many other systems will have to be devised        3 Change 50 37’30” into degrees
                                                                               0
              of an arc of length equal to radius in a circle. 0.59 degree =   0.59 x 60   = 35.4’
             of an arc of length equal to radius in a circle.
                                                                   3 Change 50 37’30” into degrees
                                                                               0
           of an arc of length equal to radius in a circle.
 (iii)   Circular System
 that is why this system is not used.
 This system is easier than Sexagesimal System.  But to     0.4 minute =   60 sec  x   0.4                          By changing angle degrees into decimals
                                                            There is one constant ratio between the circumference
                                                                      =  24”
                                                                      By changing angle degrees into decimals
                                                       1 minute
              There is one constant ratio between the circumference =   60 second
 use this system many other systems will have to be devised  Therefore 24.59° =  24 35’24”                                              30
                                                                      By changing angle degrees into decimals
             There is one constant ratio between the circumference
                                                            and dia of a circle.  This is represented by
           There is one constant ratio between the circumference and dia of a circle.  This is represented by  π . .
 (iii)   Circular System
 In this system, the unit of measuring angles is radian.  It
                                                                       =  24”
                                                                          0
              and dia of a circle.  This is represented by  π .
                                                                                          30
 that is why this system is not used.                 0.4 minute =   60 sec  x   0.4      30” =   30   = 0.50’                     30” =   60  = 0.50’
             and dia of a circle.  This is represented by  π .
 is that angle which is formed at the centre and is formed
                                                                         nce
 In this system, the unit of measuring angles is radian.  It
                                                       Therefore 24.59° =  24 35’24”
                                                               Circumfere     0
                 Circumfere
                           nce
                                                                                          60  = 0.50’
                                                                                    30” =
 of an arc of length equal to radius in a circle.
 (iii)   Circular System  Circumfere      = constant point  =  π 3 Change 50 37’30” into degrees = constant point  =  π        37’30” = 37.5’
                                                             0
                                                                =  24”
                          nce
                                                                                          60
                                                                  Diameter
                               = constant point  =  π
 is that angle which is formed at the centre and is formed
                   Diameter
                                                                                 37’30” = 37.5’
                   Diameter
                                                  3 Change 50 37’30” into degrees
                                                    By changing angle degrees into decimals
 of an arc of length equal to radius in a circle.
 There is one constant ratio between the circumference
                                                               0
 In this system, the unit of measuring angles is radian.  It  Therefore 24.59° =  24 35’24”  = π x dia = 37.5’
                                                               Circumference
                                                                     0
                                                                                37’30”
                               = π x dia
                 Circumference
 and dia of a circle.  This is represented by
                 Circumference  = π .  x dia
 is that angle which is formed at the centre and is formed  By changing angle degrees into decimals                                     37.5
                                                                        30
                               = π x dia
                Circumference
                                                                              = 2πr (where r is radius of the circle)
 There is one constant ratio between the circumference
                                                                                          37.5
                               = 2πr (where r is radius of the circle)    30” =
 of an arc of length equal to radius in a circle.  = 2πr (where r is radius of the circle)  60  = 0.50’  37.5’ =  37.5    = 0.625 0 0  37.5’ =   60    = 0.625 0
                                            3 Change 50 37’30” into degrees
                                                        0
 Circumfere
 and dia of a circle.  This is represented by
                                = 2π . r (where r is radius of the circle)
         nce
                                                                         30
                                                                                           60   = 0.625
                                                                                 22 37.5’ =
             = constant point  =  π
                                  22
                                                                          π  = 0.50’
 There is one constant ratio between the circumference  By changing angle degrees into decimals  60                         50 37’30” = 50.625 0
                                                                    30” =
                                                                              =
  Diameter
 Circumfere
                                                                         60
                                  22
          nce
                                                                                                                              0
                                                                                 7
                                                                              50 37’30” = 50.625
                                                                                0
 and dia of a circle.  This is represented by   π π  =    7    37’30” = 37.5’  50 37’30” = 50.625 0 0
               = constant point  =  π =
                                      π .   =
                                                                   30
                                                                                0
  Diameter
 Circumference
             = π x dia
                                   7
                                                                37’30” = 37.5’
                                                                       37.5
 Circumfere      = 2πr (where r is radius of the circle)      30” =   60  = 0.50’   = 0.625 0                    59                                             59
                                                                                                                  59
    nce
               = π x dia
 Circumference
                                                                37.5’ =
 Diameter  = constant point  =  π                        37’30” = 37.5’  60
                                                                         37.5
               = 2π
                22r (where r is radius of the circle)
                                                                 37.5’ =
          π
             =
                                                                          60
 Circumference  = π x dia 7 22                              50 37’30” = 50.625   = 0.625 0
                                                              0
                                                                             0
           π
        = 2πr (where r is radius of the circle)            37.5’ =  37.5   = 0.625 0 0
               =
                                                               0
                  7                                        77 50 37’30” = 50.625                59
                                                                   60
           22
     π  =                                              50 37’30” = 50.625 0                      59
                                                         0
            7
                                                                                           59
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