Warm Up
Covert to radians.
135 degrees
540 degrees
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Example 3 – Converting from Degrees to Radians
a.
Multiply by  rad / 180.
b.
Multiply by  rad / 180.
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Applications
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Applications
The radian measure formula,  = s/r, can be used to
measure arc length along a circle.
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Example 5 – Finding Arc Length
A circle has a radius of 4 inches. Find the length of the arc
intercepted by a central angle of 240, as shown in
Figure 1.12.
Figure 1.12
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Example 5 – Solution
To use the formula s = r, first convert 240 to radian
measure.
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Example 5 – Solution
cont’d
Then, using a radius of r = 4 inches, you can find the arc
length to be
s = r
Note that the units for r determine the units for r because 
is given in radian measure, which has no units.
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Your Turn
A circle has a radius of 5 inches. Find the length of the arc
intercepted by a central angle of 270, as shown in
Figure 1.12.
Figure 1.12
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Applications
The formula for the length of a circular arc can help you
analyze the motion of a particle moving at a constant speed
along a circular path.
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Example
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Linear speed
Recall the formula d = rt which holds for and object whose
average speed is known.
If we want to know the rate at which an object is moving
around a circle we can adapt this formula to be
s = vt where s is the arc length, t is the time, and v is the
angular velocity.
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Example
An object is traveling around a circle with a radius of 5 cm.
If in 20 minutes a central angle of 1/3 radians is swept out,
what is the linear speed.
We are looking for linear speed. Find s. s = r = 5(1/3)=5/3
Use s = vt where s = 5/3, t = 20 sec.
Thus 5/3 = v(20) or v = 5/60. Since v is in cm/sec we have
v = 5cm/12sec.
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Angular speed
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Applications
A sector of a circle is the region bounded by two radii of
the circle and their intercepted arc (see Figure 1.15).
Figure 1.15
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Applications
Find the area of a sector where r = 6m and theta = 120 degrees.
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