sect 1 trig
TRANSCRIPT
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Copyright 2007 Pearson Education, Inc. Slide 8-2
Trigonometric Functions And
Applications
.1 Angles and Their Measures
.2 Trigonometric Functions and FundamentalIdentities
.3 Evaluating Trigonometric Functions
.4 Applications of Right Triangles
.5 The Circular Functions
.6 Graphs of the Sine and Cosine Functions
.7 Graphs of the Other Circular Functions
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Copyright 2007 Pearson Education, Inc. Slide 8-3
.1 Angles and Arcs
Basic Terminology
Two distinct pointsA andB determine the line AB.
The portion of the line including the pointsA andB is the
line segment AB.
The portion of the line that starts atA and continues
through B is called ray AB.
An angle is formed by rotating a ray, the initial side,
around its endpoint, the vertex, to a terminal side.
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Copyright 2007 Pearson Education, Inc. Slide 8-4
Degree Measure
Developed by the Babylonians around 4000 yrs ago.
Divided the circumference of the circle into 360 parts.
One possible reason for this is because there are
approximately that number of days in a year.
There are 360 in one rotation.
An acute angle is an angle between 0 and 90.
A right angle is an angle that is exactly 90.
An obtuse angle is an angle that is greater than 90
but less than 180.
A straight angle is an angle that is exactly 180.
.1 Degree Measure
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Copyright 2007 Pearson Education, Inc. Slide 8-5
.1 Finding Measures of Complementary
and Supplementary Angles
If the sum of two positive angles is 90, the angles are
called complementary.
If the sum of two positive angles is 180, the angles are
called supplementary.
Example Find the measure of each angle in the given figure.
(a) (b)
6 3 90
10
m m
m
!
!
o
o
4 6 180
18
k k
k
!
!
o
o
(Supplementary angles)(Complementary angles)
Angles are 60 and 30 degrees Angles are 72 and 108 degrees
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Copyright 2007 Pearson Education, Inc. Slide 8-6
.1 Calculating With Degrees, Minutes,
and Seconds
One minute, written 1', is of a degree.
One second, written 1", is of a minute.
Example Perform the calculation
Solution
601
601
QQ106or1 60
1 !d!d
106or1 36001601 d!dd!!ddQ
'
.64329251 ddQQ
5783
6432
9251
d
d
d
Q
Q
Q
Since 75' = 1 + 15', the sum is written as 8415'.
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Copyright 2007 Pearson Education, Inc. Slide 8-7
.1 Converting Between Decimal Degrees
and Degrees, Minutes, and Seconds
Example
(a) Convert 748d14t to decimal degrees.
(b) Convert 34.817 to degrees, minutes, and
seconds.
Analytic Solution
(a) Since
.137.740039.1333.74
360014
6087441874
QQQQ
QQ
QQ
!}
!ddd
,1and1 36001
601 QQ !dd!d
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Copyright 2007 Pearson Education, Inc. Slide 8-8
.1 Converting Between Decimal Degrees
and Degrees, Minutes, and Seconds
(b)
Graphing Calculator Solution
2.194342.19434
)06(02.943420.9434
20.4934)06(817.34
817.34817.34
ddd!
ddd!
ddd!
dd!
d!
d!
!
Q
Q
Q
Q
Q
Q
QQQ
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Copyright 2007 Pearson Education, Inc. Slide 8-9
.1 Coterminal Angles
Quadrantal Angles are
angles in standard
position (vertex at the
origin and initial side
along the positive x-
axis) with terminal sides
along the x ory axis,
i.e. 90, 180, 270, etc.
Coterminal Angles areangles that have the same
initial side and the same
terminal side.
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Copyright 2007 Pearson Education, Inc. Slide 8-10
.1 Finding Measures of Coterminal
Angles
Example Find the angles of smallest possible positivemeasure coterminal with each angle.
(a) 908 (b) 75
Solution Add or subtract 360 as many times asneeded to get an angle between 0 and 360.
(a)
(b)
Let nbe an integer, we have an infinite number ofcoterminal angles: e.g. 60 + n 360.
QQQ 1883602908 !
QQQ
285)75(360 !
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Copyright 2007 Pearson Education, Inc. Slide 8-11
.1 Radian Measure
The radian is a real number, where the degree is a unit of
measurement.
The circumference of a circle, given by C= 2Tr, where ris the radius of the circle, shows that an angle of 360 has
measure 2T radians.
An angle with its vertex at the
center of a circle that intercepts
an arc on the circle equal in
length to the radius of the circle
has a measure of1 radian.
radians180orradians2360 TT !!QQ
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Copyright 2007 Pearson Education, Inc. Slide 8-12
.1 Converting Between Degrees and
Radians
Multiply a radian measure by 180/T and simplify
to convert to degrees. For example,
Multiply a degree measure by T /180 and simplify
to convert to radians. For example,
radians.4
radians180
4545 TT !
!Q
.405180
4
9
4
9 QQ
!
!
T
TT
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Copyright 2007 Pearson Education, Inc. Slide 8-13
.1 Converting Between Degrees and
Radians With the Graphing Calculator
Example Convert 249.8 to radians.
Solution
Put the calculator in radian mode.
Example Convert 4.25 radians to degrees.
Solution
Put the calculator in degree mode.
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Copyright 2007 Pearson Education, Inc. Slide 8-14
.1 Equivalent Angle Measures in
Degrees and Radians
Figure 18 pg 9
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Copyright 2007 Pearson Education, Inc. Slide 8-15
.1 Arc Length
Example A circle has a radius of 25 inches. Find the
length of an arc intercepted by a central angle of 45.
Solution
The length s of the arc intercepted on a circle of radius rby a
central angle of measure U radians is given by the product of
the radius and the radian measure of the angle, or
s = rU, U in radians
inches25.64
25
radians4180
4545
TT
TT
!
!
!
!
s
Q
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Copyright 2007 Pearson Education, Inc. Slide 8-16
.1 Linear and Angular Speed
Angular speed [(omega) measures the speed of
rotation and is defined by
Linear speed Y is defined by
Since the distance traveled along a circle is given
by the arc length s, we can rewrite Y as
.in timeradiansin, tt
UU
[ !
.in timedistancelinearis, tst
s!Y
.or, [YUU
Y rt
rt
r
t
s!
!!!
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Copyright 2007 Pearson Education, Inc. Slide 8-17
.1 Finding Linear Speed and Distance
Traveled by a Satellite
Example A satellite traveling in a circular orbit 1600 km
above the surface of the Earth takes two hours to complete an
orbit. The radius of the Earth is 6400 km.
(a) Find the linear speed of the satellite.
(b) Find the distance traveled in 4.5 hours.
Figure 24 pg 12
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Copyright 2007 Pearson Education, Inc. Slide 8-18
.1 Finding Linear Speed and Distance
Traveled by a Satellite
Solution
(a) The distance from the Earths center is
r= 1600 + 6400 = 8000 km.
For one orbit, U = 2T, so s = rU = 8000(2T) km. Witht= 2 hours, we have
(b) s = Yt= 8000T(4.5) } 110,000 km
kph.000,258000
2
)2(8000}!!! T
TY
t
s