Transcript 3.3 - James Bac Dang

3.3 Definition III: Circular Functions
• A unit circle has its center at the origin and
a radius of 1 unit.
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Circular Functions
• sin s  y
1
csc s  ( y  0)
y
cos s  x
y
tan s  ( x  0)
x
1
sec s  ( x  0)
x
x
cot s  ( y  0)
y
2
Unit Circle
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Domains of the Circular Functions
• Assume that n is any integer and s is a real
number.
• Sine and Cosine Functions: (, )

 

• Tangent and Secant Functions: s | s   2n  1  
2 


• Cotangent and Cosecant Functions: s | s  n 
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Evaluating a Circular Function
• Circular function values of real numbers
are obtained in the same manner as
trigonometric function values of angles
measured in radians.
• This applies both to methods of finding
exact values (such as reference angle
analysis) and to calculator approximations.
Calculators must be in radian mode when
finding circular function values.
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Example: Finding Exact Circular Function Values
7
7
7
• Find the exact values of sin , cos , and tan .
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4
4
• Evaluating a circular function at the real number 7
7
4
is equivalent to evaluating it at 4 radians. An
7
angle of 4 intersects the unit circle at the point
 2  2 .
,

 2

2 
• Since sin s = y, cos s = x, and
7  2
• sin

4
2
7
2
cos

4
2
y
tan s 
x
7  22
tan
 2  1
4
2
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Example: Approximating
• Find a calculator approximation to four decimal places
for each circular function. (Make sure the calculator is in
radian mode.)
• a) cos 2.01  .4252
b) cos .6207  .8135
– For the cotangent, secant, and cosecant functions
values, we must use the appropriate reciprocal
functions.
• c) cot 1.2071
1
cot1.2071 
 .3806
tan1.2071
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3.4 Arc Length and Area of a Sector
• The length s of the
arc intercepted on a
circle of radius r by a
central angle of
measure  radians is
given by the product
of the radius and the
radian measure of the
angle, or
s = r,  in radians.
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Example: Finding Arc Length
• A circle has radius 18.2
cm. Find the length of the
arc intercepted by a
central angle having each
of the following measures.
3
• a)
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• b) 144
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Example: Finding Arc Length -- continued
• a) r = 18.2 cm and  = 3
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• s  r
 3 
s  18.2   cm
 8 
54.6
s
cm  21.4cm
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• b) convert 144 to radians
  
144  144 

180


4

radians
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s  r
 4 
s  18.2 
 cm
 5 
72.8
s
cm  45.7cm
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Example: Finding a Length
• A rope is being wound
around a drum with
radius .8725 ft. How
much rope will be wound
around the drum it the
drum is rotated through
an angle of 39.72?
• Convert 39.72 to radian
measure.
s  r

  
s  .8725 39.72 
   .6049 ft.
 180  

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Example: Finding an Angle
Measure
• Two gears are
adjusted so that the
smaller gear drives
the larger one, as
shown. If the smaller
gear rotates through
225, through how
many degrees will the
larger gear rotate?
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Solution
• Find the radian measure of the angle and then find the
arc length on the smaller gear that determines the
motion of the larger gear.
   5
225  225 

 180  4
 5  12.5 25
s  r  2.5 

cm.

4
8
 4 
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Solution continued
• An arc with this length on the larger gear
corresponds to an angle measure , in
s  r
radians where
25
 4.8
8
125

192
125  180 

  117
192   
• Convert back to degrees.
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Area of a Sector
• A sector of a circle is a portion of the
interior of a circle intercepted by a central
angle. “A piece of pie.”
• The area of a sector of a circle of radius r
and central angle  is given by
1 2
A  r ,
2
 in radians.
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Example: Area
• Find the area of a sector with radius 12.7 cm
and angle  = 74.
• Convert 74 to radians.
  
74  74 
  1.292 radians
 180 
• Use the formula to find the area of the sector of
a circle.
1 2
1
A  r   (12.7)21.292  104.193 cm 2
2
2
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