Mat 161 PreCalculus

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Mat 161 - PreCalculus

• Section 5.1
• Angles and Their Measure

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Angles and Their Measure

• Def An angle is formed by the joining of two
  rays at their endpoints.

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Angles and Their Measure

• We measure angles in degrees or radians. The
  measure of an angle is the size of the rotation
  from the initial side to the terminal side of the
  angle.
• We use either radian or degree measure for the
  purposes of this course when we measure angles.

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Angles and Their Measure

• We may recall that a complete rotation, which
  is formed when we rotate a ray in a complete
  cirle, has a measure of 360 degrees, or 360.
• So, we say 1 revolution 360.

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Angles and Their Measure

• How do we define radian measure?
• Def Given a circle of radiuc r. Let s be the
  length of an arc on this circle, then the measure
  of the central angle ? that intercepts the arc is
  
• ? s/r radians

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Angles and Their Measure

• Example
• Find the radian measure of the central angle
  of a circle of radius 6 feet that intercepts an
  arc of length 15 feet.

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Angles and Their Measure

• So, in particular, if we consider a circle of
  radius r unit and consider the arc which is 1
  complete revolution then what is the radian
  measure of the angle that is the central angle of
  the circle?
• ? s/r

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Angles and Their Measure

• Thus, the conversion rule is
• 360 2p radian
• This means that
• 1 2p/360 p/180 radian
• 1 radian 360/2p

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Angles and Their Measure

• Examples
• Convert the following angles in degrees to
  radian measure.
• a) 20 e) 60
• b) 180 f) 30
• c) -45 g) 90
• d) 720 h) -720

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Angles and Their Measure

• Examples
• Convert the following angles in radians to
  degree measure.
• a) p
• b) p/18
• c) -p/20
• d) 3

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Angles and Their Measure

• To place an angle in standard position, we
• 1) draw a cartesian coordinate system,
• 2) draw a unit circle,
• 3) draw the initial side of the angle on the
  positive x-axis by placing its vertex on the
  origin and then
• 4) rotating in CW or CCW direction draw the
  terminal side of the angle and represent the
  angle with an arrow in the proper direction.
• (Note the point where the terminal side of
  the angle intersects the unit circle we shall
  call the TERMINAL POINT.)

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Angles and Their Measure

• Example Draw 90 in standard position.
• And, then identify the TP.

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Angles and Their Measure

• Example Draw -180 in standard position. And,
  then identify the TP.

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Angles and Their Measure

• Example Draw 360º in standard position.
• And, then identify the TP.

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Angles and Their Measure

• Note
• Two angles are coterminal if their terminal sides
  are the same. The difference between these angles
  is some multiple of 360. (List some ?)
• Two positive angles are complements of each other
  if their sum is 90. (List some ?)
• Two positive angles are supplements of each other
  if their sum is 180. (List some ?)

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Angles and Their Measure

• Question
• How would you find the length of the arc on a
  circle of radius 20 inches that is intercepted by
  a central angle of 315?

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MAT 161

• References
• Algebra and Trigonometry by Blitzer
• Third Edition