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Mathematics Dictionary
Dr. K. G. Shih
Angle and Sectors
Subjects

  • TR 01 00 | - Outline
  • TR 01 01 | - Unit of angle
  • TR 01 02 | - Arc length of a sector
  • TR 01 03 | - Area of a sector
  • TR 01 04 | - Application of arc length : Find height of tree
  • TR 01 05 | - Measurement of angles in trigonometry
  • TR 01 06 | - Angle and quadrants
  • TR 01 07 | -
  • TR 01 08 | -
  • TR 01 09 | -
  • TR 01 10 | -
    • Answers


    TR 01 01. Unit of angle

    • English method or Sexagesimal method
      • 1° = (Right angle)/90. This is defintion of degree.
      • 1' = 1°/60. This is one minute.
      • 1" = 1'/60. This is one second
    • French method or Centesimal method
      • 1 grade = (Right angle)/100.
      • 1 minute = 1 grade/100.
      • 1 second = 1 minute/60.
    • Radians
      • Definition
        • A circle arounds its center has angle of 2*pi radians.
        • Right angle has pi/2 radians.
    • Relation between degree and radian
      • One radian = 360 degree/(2*pi) = 57.29579 degrees.
      • One degree = pi/180 = 0.017453 radian

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    TR 01 02. Arc length of a sector

    Diagrams
    • Cobstruction
      • Draw a circle with center 0 and radius r.
      • Draw two points A and B on circle. Then OA = OB = r = radius of circle.
      • OA, OB and arc AB formed a sector.
    • Find arc AB
      • Formula : s = r*A
      • Where s is length of arc, r is radius and A is angle of sectors.
      • Angle A is in radians
    • Example : Sector ahs radius 10 cm and angle 30 degrees. Find length of arc.
      • Angle of sector is 30 degrees = pi/6 radians.
      • Length of arc of sector = 10*pi/6 cm.

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    TR 01 03. Area of a sector

    Diagrams     Defintion
    • Formula : Area = (r^2)*A/2.
    • Where r is radius and A is angle in radians.
    Example : Find area of a sector whose angle is 45 degrees and radius is 10 cm.
    • Angle of sector = 45 degrees = pi/4 radians.
    • Hence area of sector = ((10)^2)*(angle of sector)/2 = (100*pi/4)/2 = 12.5*pi.
    Example : If Angle A = 2*pi, the area = pi*r^2 is the area of circle

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    TR 01 04. Application of arc length

    Distance from tree is 100 meters and subtend angle of tree is 2 degrees.
    • d = 100 ft.
    • A = 2 degrees = 2*pi/180 = 0.034906 radians.
    • Assume radius r = d = 100 m.
    • Assume height of tree is arc of sector.
    • Hence height of tree is h = r*A = 100*0.034906 = 3.4 meters.

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    TR 01 05. Measurement of angles in trigonometry

    Diagrams
    • Diagrams Dirction of angle measurement in diagram 01 01
    Measurements
    • Measure angle anti-clockwise from x-axis OX give positive angle.
    • Measure angle clockwise from x-axis OX give negative angle.
    Angle POX
    • Angle POX = 090 degrees. OP is perpendicular to OX and P is above O.
    • Angle POX = 180 degrees. X,O,P are co-linear and P is at left side of O.
    • Angle POX = 270 degrees. OP is perpendicular to OX and P is below O.
    • Angle POX = 360 degrees. OP and OX are coincided.
    • Angle POX = 450 degrees. OP is perpendicular to OX and P is above O.
    • Angle POX = 720 degrees. OP and OX are coincided.
    Note
    • Angle in geometry : Only consider angles less than 360 degrees
    • Angle in trigonometry : Angles can be geater than 360 degrees

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    Q06. Angle and quadrants

    First quadrant
    • Angle between 0 and 90 degrees
    • x is positive
    • y is positive
    Second quadrant
    • Angle between 90 and 180 degrees
    • x is negative
    • y is positive
    Third quadrant
    • Angle between 180 and 270 degrees
    • x is negative
    • y is negative
    Fourth quadrant
    • Angle between 270 and 360 degrees
    • x is positive
    • y is negative

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    TR 01 07.


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    TR 01 08.


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    TR 01 09. Answer


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    TR 01 10. Answer


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    Q00. Outline
    Unit of angle (TR 01 01)
    • One radian = 180/pi = 57.2956 degrees
    • One degree = pi/180 = 0.017453 radians
    Arc length of sector (Tr 01 02)
    • S = r*A
    • Where r is radius and A is angle in radians
    Area of sector (Tr 01 03)
    • Area = (r^2)*A/2
    • Where r is radius and A is angle in radians
    Circle
    • Circumference = 2*pi*r
    • Ares = pi*(r^2)
    Quadrants
    • 1st quadrant : Angle = 000 to 090 degrees. x = (+) and y = (+)
    • 2nd quadrant : Angle = 090 to 180 degrees. x = (-) and y = (+)
    • 3rd quadrant : Angle = 180 to 270 degrees. x = (-) and y = (-)
    • 4th quadrant : Angle = 270 to 360 degrees. x = (+) and y = (-)

    Go to Begin
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