Section 4.1 Radian and Degree Measure

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Section 4.1 Radian and Degree Measure

• Pages 282 - 293

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What you should learn

• Describe angles.
• Use radian measure.
• Use degree measure.
• Use angles to model and solve real-life
  problems.

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An angle is determined by rotating a ray
(half-line) about its endpoint. The starting
position of the ray is the initial side of the
angle, and the position after rotation is the
terminal side, as shown in Figure 1
An angle is in standard position, as shown in
Figure 2.
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Positive angles are generated by counterclock
wise rotation, and negative angles by clockwise
rotation, as shown in Figure 3.
Angles are labeled with Greek letters like alpha,
beta, and theta, as well as uppercase letters.
Figure 4, shows angles that are coterminal.
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• Definition of Radian
• One radian is the measure of a central angle
  ? that intercepts an arc (s) equal in length to
  the radius (r) of the circle. See Figure 5.
• Algebraically, this means that ,
  where ? is measured in radians
• 
  

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These are some common angles (must know)
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Example 1 Sketching and Finding Coterminal Angles

a) For the positive angle 13p/6, subtract 2 p to
obtain a coterminal angle.
b) For the positive angle 3p/4, subtract 2 p to
obtain a coterminal angle
c) For the negative angle -2p/3, add 2 p to
obtain a coterminal angle
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Complementary Angles
Two angles are complementary angles if their
sum equal p/2
Supplementary Angles
Two angles are supplementary angles if their
sum equal p
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Example 2 Complementary and Supplementary Angles
If possible, find the complement and the
supplement of a) 2p/5 b) 4p/5
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Example 3 Converting from Degrees to Radians
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Example 4 Converting from Radians to Degrees
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Applications
The radian measure formula, ? s/r, can be used
to measure the arc length along a circle.
Arc Length For a circle of radius r, a central
angle ? intercepts an arc of length s given by
s r? Where ? is measured in radians, Note that
if r 1, then s ?, and the radian measure of ?
equals the arc length.
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Example 5 Finding Arc Length A circle has radius
of 4 inches. Find the length of the arc
intercepted by a central angle of 2400. As shown
in the figure.
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Linear and Angular Speeds
Consider a particle moving at a constant speed
along a circular arc with radius r. If s is the
length of the arc traveled in t time, then the
linear speed v of the particle is Linear speed
Moreover, if ? is the angle (in radian
measure) corresponding to the arc length s, then
the angular speed ? (the lower case Greek letter
omega) of the particle is Angular speed
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Example 6 Finding Linear Speed The second hand
of a clock is 10.2 centimeters long, as shown in
the figure. Find the linear speed of the tip of
this second hand as it passes around the clock
face.
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Example 7 Finding Angular and Linear Speeds A
Ferris wheel with a 50-foot radius makes 1.5
revolutions per minute. a) Find the angular speed
of the Ferris wheel in radians per minute. b)
Find the linear speed of the Ferris wheel.
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Area of a Sector of a Circle
For a circle of radius r, the area A of a sector
of the circle with central angle ? measured in
radians is given by
If the angle is in degrees, the following formula
can be used
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Example 8 Area of a Sector of a Circle A
sprinkler on a golf course fairway is set to
spray water over a distance of 70 feet and
rotates through an angle of 1200. Find the area
of the fairway watered by the sprinkler.