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May 28 Math 1113 sec 002 Summer 2014
Arclength Formula (sec. 6.1)
Given a circle of radius r , the length s of the arc subtended by the
(positive) central angle θ (in radians) is given by
s = r θ.
The area of the resulting sector is A = 12 r 2 θ.
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Motion on a Circle: Angular & Linear Speed
Definition: (angular speed) If an object moves along the arc of a
circle through a central angle θ in the time t, the angular speed is
denoted by ω (lower case omega) and is defined by
θ
ω= .
t
Definition: (linear speed) If the circle has radius r , then the distance
traveled is the arclength s = r θ. The linear speed is denoted by ν
(lower case nu) and is defined by
ν=
()
s
rθ
=
= r ω.
t
t
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Example
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Example
A child is sitting on the edge of a merry-go-round that makes three full
revolutions per minute. If the radius of the merry-go-round is 6 ft, what
is her linear speed.
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Caveat!
Remember that the formulas for
arclength, sector area, angular speed, & linear
speed
are for an angle in radians. An angle in degrees
must be converted to radians before applying any
of these formulas.
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Section 6.2: Trigonometric Functions of Acute Angles
In this section, we are going to define six new functions called
trigonometric functions. We begin with an acute angle θ in a right
triangle with the sides labeled:
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Sine, Cosine, and Tangent
For the acute angle θ, we define the three numbers as follows
sin θ =
cos θ =
tan θ =
opp
,
hyp
,
hyp
opp
,
Note that these are numbers, ratios of side lengths, and have no units.
It may be convenient to enclose the argument of a trig function in
parentheses. That is,
sin θ = sin(θ).
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Cosecant, Secant, and Cotangent
The remaining three trigonometric functions are the reciprocals of the
first three
csc θ =
1
hyp
=
,
opp
sin θ
sec θ =
cot θ =
()
hyp
1
=
,
cos θ
1
=
,
opp
tan θ
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Example
Determine the six trigonometric values of the acute angle θ.
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Example
Determine the six trigonometric values of the acute angle θ.
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Some Key Trigonometric Values
Use the triangles to determine the six trigonometric values of the
angles π4 , π3 and π6 .
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Commit To Memory
It is to our advantage to remember the following:
sin π6 = 12 ,
cos π6 =
tan π6 =
√
3
2 ,
√1 ,
3
√
sin π4 =
√1 ,
2
sin π3 =
cos π4 =
√1 ,
2
cos π3 =
tan π4 = 1,
tan π3 =
3
2
1
2
√
3
We’ll use these to find some other trigonometric values. Still others will
require a calculator.
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Calculator
Figure: Remember to check the mode for radians or degrees. (TI-84 shown)
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Using a Calculator
Evaluate the following using a calculator. Round answers to three
decimal places.
sin
π
=
8
sec 78.3◦ =
tan(0.2) =
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Application Example
Before cutting down a dead tree, you wish to determine its height.
From a horizontal distance of 40 ft, you measure the angle of elevation
from the ground to the top of the tree to be 61◦ . Determine the tree
height to the nearest 100th of a foot.
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Application Example
The ramp of a moving truck touches the ground 14 feet from the end of
the truck. If the ramp makes an angle of 28.5◦ with the ground, what is
the length of the ramp?
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