4.1 Radian and Degree Measure

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4.1 Radian and Degree Measure

• Trigonometry- from the Greek measurement of
  triangles
• Deals with relationships among sides and angles
  of triangles and is used in astronomy,
  navigation, and surveying.

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Angles

• An angle is two rays with the same initial
  point.
• The measure of an angle is the amount of
  rotation required to rotate one side, called the
  initial side, to the other side, called the
  terminal side.
• The shared initial point of the two rays is
  called the vertex of the angle.

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Angles in standard position

• Standard position - the vertex is at the origin
  of the rectangular coordinate system and the
  initial side lies along the positive x-axis.
• If the rotation of the angle is in the
  counterclockwise direction, then the angle is
  said to be positive. If the rotation is
  clockwise, then the angle is negative.

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Coterminal Angles

• Two angles in standard position that have the
  same terminal side are said to be coterminal.

You find coterminal angles by adding or
subtracting
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Radians vs. Degrees

• One radian is the central angle required to
  stretch the radius around the outside of the
  circle.
• Since the circumference of a circle is
  , it takes
• radians to get completely around the circle
  once. Therefore, it takes radians to get
  halfway around the circle.

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Common Radian Angles, pg. 285
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Angles between 0 and are acute Angles
between and p are obtuse
p/2
p/2
? p/2
Quad II Quad I
p/2 lt ? lt p 0 lt ? lt p/2
? p
? 0, Quad III
Quad IV 2p p lt ? lt 3p/2
3p/2 lt ? lt 2p
? 3p/2
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Types of angles

• Complementary Angles
• Angles that add up to or
• Supplementary Angles
• Angles that add up to or

?
ß
ß
?
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One degree is equivalent to a rotation of 1/360
of a complete revolution about the vertex.
To convert degrees to radians, multiply the
degrees by To convert radians to
degrees multiply the radians by
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• Convert the following degree measures to radian
  measure.
• a) 120
• -315
• 12

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• Convert the following radian measures to degrees.
• 5p/6
• 7

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Arc Length For a circle of radius r, a central
angle ? intercepts an arc of length s given by
sr?, where ? is measured in radians
Example Find the length of the arc that subtends
a central angle with measure 120 in a circle
with radius 5 inches
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Angular and Linear Velocity

• Angular Velocity (?) is the speed at which
  something rotates. Therefore,
  which means the rotation per unit time (how fast
  something is going around a circle).
• Linear Velocity (v) is the speed at which the
  outside tip of the radius is traveling.
  Therefore, v r?. This equation considers the
  number of radii (since is expressed in
  radians) that travel around the circle during the
  rotation process.

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Example 1 A lawn roller with a 10-inch radius
makes 1.2 revolutions per second. a.) Find the
angular speed of the roller in radians per
second. b.) Find the speed of the tractor that
is pulling the roller in mi/hr.
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Example 2 The second hand of a clock is 10.2 cm
long. Find the linear speed of the tip of this
hand.
Example 3 An automobile is traveling at 65 mph.
If each tire has a radius of 15 inches, at what
rate are the tires spinning in revolutions per
minute (rpm)?
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Assignment

• Page 291 5-21 odd, 35-65 odd, 75-85 odd, 95, 97