Презентация по математике на тему "Radian and Degree Measure"

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  6.1 Radian and Degree Measure
  In this section, we will study the following topics:
  
  Terminology used to describe angles
  Degree measure of an angle
  Radian measure of an angle
  Converting between radian and degree measure
  Find coterminal angles
  
  
  

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  6.1 Radian and Degree Measure
  Angles
  Trigonometry: measurement of triangles
  
  Angle Measure
  
  
  

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  6.1 Radian and Degree Measure
  Standard Position
  Vertex at origin
  The initial side of an angle in standard position is always located on the positive x-axis.
  

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  6.1 Radian and Degree Measure
  Positive and negative angles
  When sketching angles, always use an arrow to show direction.
  

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  6.1 Radian and Degree Measure
  Measuring Angles
  The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.
  
  There are two common ways to measure angles, in degrees and in radians.
  We’ll start with degrees, denoted by the symbol º.
  One degree (1º) is equivalent to a rotation of of one revolution.
  
  

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  6.1 Radian and Degree Measure
  Measuring Angles
  
  
  
  
  
  
  
  

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  Angles are often classified according to the quadrant in which their terminal sides lie.
  
  Ex1: Name the quadrant in which each angle lies.
  50º
  208º II I
  -75º III IV
  
  6.1 Radian and Degree Measure
  Classifying Angles
  Quadrant 1
  Quadrant 3
  Quadrant 4
  

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  6.1 Radian and Degree Measure
  Classifying Angles
  Standard position angles that have their terminal side on one of the axes are called quadrantal angles.
  
  For example, 0º, 90º, 180º, 270º, 360º, … are quadrantal angles.
  

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  6.1 Radian and Degree Measure
  Coterminal Angles
  Angles that have the same initial and terminal sides are coterminal.
  
  Angles  and  are coterminal.
  

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  6.1 Radian and Degree Measure
  Example of Finding Coterminal Angles
  You can find an angle that is coterminal to a given angle  by adding or subtracting multiples of 360º.
  
  Ex 2:
  Find one positive and one negative angle that are coterminal to 112º.
  For a positive coterminal angle, add 360º : 112º + 360º = 472º
  For a negative coterminal angle, subtract 360º: 112º - 360º = -248º
  
  

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  Ex 3. Find one positive and one negative angle that is coterminal with the angle  = 30° in standard position.
  
  Ex 4. Find one positive and one negative angle that is coterminal with the angle  = 272 in standard position.
  
  

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  6.1 Radian and Degree Measure
  Radian Measure
  A second way to measure angles is in radians.
  
  Definition of Radian:
  One radian is the measure of a central angle  that intercepts arc s equal in length to the radius r of the circle.
  
  
  In general,
  
  
  
  

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  6.1 Radian and Degree Measure
  Radian Measure
  

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  6.1 Radian and Degree Measure
  Radian Measure
  

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  6.1 Radian and Degree Measure
  Conversions Between Degrees and Radians
  To convert degrees to radians, multiply degrees by
  
  To convert radians to degrees, multiply radians by
  

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  Ex 5. Convert the degrees to radian measure.
  
  60
  
  30
  
  -54
  
  -118
  
  45
  

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  Ex 6. Convert the radians to degrees.
  a)
  
  b)
  
  c)
  
  d)
  

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  Ex 7. Find one positive and one negative angle that is coterminal with the angle  = in standard position.
  
  Ex 8. Find one positive and one negative angle that is coterminal with the angle  = in standard position.
  
  

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  0° 
  360 ° 
  30 ° 
  45 ° 
  60 ° 
  330 ° 
  315 ° 
  300 ° 
   120 °
   135 °
   150 °
   240 °
   225 °
   210 °
   180 °
  90 ° 
  270 ° 
  
  Degree and Radian Form of “Special” Angles
  

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  Class Work
  Convert from degrees to radians.
  54
  -300
  
  Convert from radians to degrees.
  3.
  
  4.
  

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  Find one postive angle and one negative angle in standard position that are coterminal with the given angle.
  135