Transcript Section 4.3 PowerPoint

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Slide 4- 1
Chapter 4
Trigonometric Functions
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4.1
Angles and Their Measures
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Quick Review
1. Find the circumference of the circle with a radius of 4.5 in.
2. Find the radius of the circle with a circumference of 14 cm.
3. Given s  r . Find s if r  2.2 cm and   4 radians.
4. Convert 65 miles per hour into feet per second.
5. Convert 9.8 feet per second to miles per hour.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 4
Quick Review Solutions
1. Find the circumference of the circle with a radius of 4.5 in.
9 in
2. Find the radius of the circle with a circumference of 14 cm. 7 /  cm
3. Given s  r . Find s if r  2.2 cm and   4 radians. 8.8 cm
4. Convert 65 miles per hour into feet per second. 95.3 feet per second
5. Convert 9.8 feet per second to miles per hour.
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6.681 miles per hour
What you’ll learn about




The Problem of Angular Measure
Degrees and Radians
Circular Arc Length
Angular and Linear Motion
… and why
Angles are the domain elements of the trigonometric
functions.
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HIPPARCHUS OF NICAEA (190–120 B.C.)
Hipparchus of Nicaea, the “father of trigonometry,” compiled
the first trigonometric tables to simplify the study of astronomy
more than 2000 years ago. Today, that same mathematics
enables us to store sound waves digitally on a compact
disc. Hipparchus wrote during the second century B.C., but he
was not the first mathematician to “do” trigonometry. Greek
mathematicians like Hippocrates of Chois (470–410 B.C.) and
Eratosthenes of Cyrene (276–194 B.C.) had paved the way
for using triangle ratios in astronomy, and those same triangle
ratios had been used by Egyptian and Babylonian engineers
at least 4000 years earlier. The term “trigonometry” itself
emerged in the 16th century, although it derives from ancient
Greek roots: “tri” (three), “gonos” (side), and “metros”
measure).
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Why 360°?
The idea of dividing a circle into 360 equal pieces dates
back to the sexagesimal (60-based) counting system of the
ancient Sumerians. The appeal of 60 was that it was
evenly divisible by so many numbers (2, 3, 4, 5, 6, 10, 12,
15, 20, and 30).
The Sumerian civilization was believed to have started
around 4000 BC. They pretty much disappeared around
2000 BC, mostly due to war between them and other
semitic people groups.
Early astronomical calculations wedded the sexagesimal
system to circles, and the rest is history.
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Slide 4- 8
Angle to linear measure
The ratio of s to h in the right triangle in Figure 4.1a is independent of the size of
the triangle. (You may recall this fact about similar triangles from geometry.) This
valuable insight enabled early engineers to compute triangle ratios on a small
scale before applying them to much larger projects. That was (and still is)
trigonometry in its most basic form.
For astronomers tracking celestial motion, however, the extended diagram in
Figure 4.1b was more interesting. In this picture, s is half a chord in a circle of
radius h, and  is a central angle of the circle intercepting a circular arc of
length a. If  were 40 degrees, we might call a a “40-degree arc” because of
its direct association with the central angle , but notice that a also has a length
that can be measured in the same units as the other lengths in the picture. Over
time it became natural to think of the angle being determined by the arc rather
than the arc being determined by the angle, and that led to radian measure.
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Degrees, Minutes, and Seconds



a degree, ° is equal to 1/360th of a circle or
1/180th of a straight angle.
a minute, ′, is 1/60th of a degree.
a second, ″, is 1/60th of a minute or 1/3600th of
a degree.
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Working with DMS Measure
(a) Convert 42º 24′ 36″ to degrees.
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Working with DMS Measure
(b) Convert 37.425º to DMS.
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Slide 4- 12
Working with DMS Measure-Calculator
(b) Convert 37.425º to DMS.
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Slide 4- 13
Radian
A central angle of a circle has measure 1
radian if it intercepts an arc with the same
length as the radius.
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Slide 4- 14
2  radians = 1 circle = 360
 radians = ½ circle = 180
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Slide 4- 15
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Slide 4- 16
Degree-Radian Conversion
180
To convert radians to degrees, multiply by
.
 radians
To convert degrees to radians, multiply by
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 radians
180
.
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Happy Tau Day!
Thu, 28 Jun 2012 14:16:08
You may have heard of (or even celebrated) Pi day on 3/14, but how
about Tau day? What is Tau, you ask? In 2001, Bob Palais published
the article "π Is Wrong" in which he argued that the beloved
constant π is the wrong choice of circle constant. He instead
proposed using an alternate constant equal to 2π, or 6.283… to
represent “1 turn”, so that 90 degrees is equal to “a quarter turn”,
rather than the seemingly arbitrary “one-half π”. Two years ago
today, Michael Hartl published "The Tau Manifesto" echoing the
good points made by Palais and building on them by calling this “1
turn” constant τ (tau), as an alternative to π. Tau is defined as the
ratio of a circle’s circumference to its radius, not its diameter and is
equal to 2π.
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Slide 4- 18
90 degrees = how many radians?
90

3

180


2
radians
radians equals how many degrees?
 180
 60
3 
or
180
 60 Replace π with 180°.
3
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Slide 4- 19
Example Working with Radian Measure
How many radians are in 60 degrees?
Since  radians and 180 both measure a straight angle, use the conversion
factor  radians  / 180
  1 to convert radians to degrees.

  radians  60
60 
radians  radians

3
 180
 180
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Slide 4- 20
Arc Length Formula (Radian Measure)
Since a central angle of 1 radian always intercepts an arc of one
radius in length, it follows that a central angle of  radians in a
circle of radius r intercepts an arc of length  r.
This gives us a convenient formula for measuring arc length.
If  is a central angle in a circle of radius r , and if  is measured in
radians, then the length s of the intercepted arc is given by
s  r .
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Slide 4- 21
Arc Length Formula (Degree Measure)
If  is a central angle in a circle of radius r , and if  is measured in
degrees, then the length s of the intercepted arc is given by
 r
s
. (convert  to radians and multiply by the radius, r )
180
Arc Length Formula (Degree Measure)
If is a central angle in a circle of radius r, and if is measured in degrees,
then the length s of the intercepted arc is given by
  
s  r  d 

180


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Slide 4- 22
Example Perimeter of a Pizza Slice
Find the perimeter of a 30 slice of a large 8 in. radius pizza.
Let s equal the arc length of the pizza's curved edge.
  8  30 
240
s

 4.2 in.
180
180
P  8 in.  8 in.  s in.
P  20.2 in.
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Slide 4- 23
Designing a Running Track
The running lanes at the Emery Sears track at Bluffton
College are 1 meter wide. The inside radius of lane 1 is
33 meters and the inside radius of lane 2 is 34 meters.
How much longer is lane 2 than lane 1 around one turn?
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Slide 4- 24
SOLUTION
We think this solution through in radians. Each lane is a
semicircle with central angle  = , and length
s = r  = r . The difference in their lengths, therefore,
is 34 - 33   . Lane 2 is about 3.14 meters longer
than lane 1.
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Slide 4- 25
Angular and Linear Motion


Angular speed is measured in units like
revolutions per minute.
Linear speed is measured in units like miles
per hour.
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Slide 4- 26
Using Angular Speed
Albert Juarez’s truck has wheels 36 inches in diameter.
If the wheels are rotating at 630 rpm (revolutions per
minute), find the truck’s speed in miles per hour.
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Slide 4- 27
Solution
We convert revolutions per minute to miles per hour by a
series of unit conversion factors. Note that the conversion
factor 18in/1 radian works for this example because the
radius is 18 in.
630 rev 60 min 2 rad 18 in 1 ft
1 mi
1 min
1 hr
1 rev 1 rad 12 in 5280 ft
= 67.47 mph
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Slide 4- 28
Navigation
In navigation, the course or
bearing of an object is
sometimes given as the
angle of the line of travel
measured clockwise from
due north.
The boat’s bearing is 155
degrees.
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Slide 4- 29
Nautical Mile
A nautical mile (naut mi, NM, nmi) is the length
of 1 minute of arc along Earth’s equator.
Traveling one nautical mile per hour is known
as a knot.
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Slide 4- 30
How big is an nautical mile?





The diameter of the earth is about 7912.18 statute
(land) miles.
The radius at the equator is approximately 3956 statute
miles.
1′ is 1/60th of a degree.
 
 1 
1'   
 60 
180


10,800
radians
so 1 nautical mile = 3956 (/10,800) = 1.151 statute
miles.
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Slide 4- 31
Distance Conversions
1 statute mile  0.87 nautical mile
1 nautical mile  1.15 statute mile
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Slide 4- 32
hwk pg 358, #1-47 odd, 53-56
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Slide 4- 33
4.2
Trigonometric Functions of Acute
Angles
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