Review and Section 5.1
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Table of Contents
1
Review
2
Section 5.1
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Review Topics
Important things to remember:
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Review Topics
Important things to remember:
How to simplify radicals.
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Review Topics
Important things to remember:
How to simplify radicals.
How to solve quadratics.
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Review Topics
Important things to remember:
How to simplify radicals.
How to solve quadratics.
How to simplify fractions.
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Review Topics
Important things to remember:
How to simplify radicals.
How to solve quadratics.
How to simplify fractions.
How to calculate area/circumference of circles.
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Simplifying Radicals
Example: Simplify
√
198.
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Simplifying Radicals
r
Example: Simplify
2
. Don’t forget to rationalize the denominator!
5
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Simplifying Radicals
√
√
Example: Simplify: 4 7 + 28.
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Simplifying Radicals
Example: Simplify
√
6·
√
10.
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Solving Quadratics
Example: Solve the quadratic equation (x + 2)(x − 7) + 20 = 0.
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Simplifying Fractions
Example: Simplify
1 − 53
.
7
6 +1
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Area and Circumference of a Circle
Example: Find the area and circumference of a circle with a diameter of 8
cm.
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Measure of Angles
We measure angles in two ways:
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Measure of Angles
We measure angles in two ways:
In degrees
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Measure of Angles
We measure angles in two ways:
In degrees
In radians
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Degrees
If we consider a ray we can then rotate that ray about its endpoint. Some
notes about that rotation
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Degrees
If we consider a ray we can then rotate that ray about its endpoint. Some
notes about that rotation
One full rotation is 360◦ .
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Degrees
If we consider a ray we can then rotate that ray about its endpoint. Some
notes about that rotation
One full rotation is 360◦ .
1◦ is
1
360
of a rotation.
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Degree Minutes and Seconds
We can further break down a degree into smaller intervals of measure
called minutes and seconds.
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Degree Minutes and Seconds
We can further break down a degree into smaller intervals of measure
called minutes and seconds.
1 minute =
1 ◦
60
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Degree Minutes and Seconds
We can further break down a degree into smaller intervals of measure
called minutes and seconds.
1 minute =
1 second =
1 ◦
60
1
60 minutes
=
1 ◦
3600
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Degrees Minutes and Seconds
Example: Convert 226.789◦ into degree-minute-second form.
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Degrees Minutes and Seconds
Example: Convert 10◦ 530 47” into decimal degrees.
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Radian Measure
We can also measure a central angle by taking the ratio of the arc length
and radius of the circle. That is
θ=
s
r
.
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Radian Measure
How many radians is one full rotation?
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Radian Measure
How many radians is one full rotation?
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Radian Measure
How many radians is one full rotation?
θ=
s
r
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Radian Measure
How many radians is one full rotation?
s
r
circumference
=
radius
θ=
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Radian Measure
How many radians is one full rotation?
s
r
circumference
=
radius
2πr
=
r
θ=
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Radian Measure
How many radians is one full rotation?
s
r
circumference
=
radius
2πr
=
r
= 2π
θ=
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Radians and Degrees
What relationships exist between degrees and radians?
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Radians and Degrees
What relationships exist between degrees and radians?
If 2π = 360◦ , then π =
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Radians and Degrees
What relationships exist between degrees and radians?
If 2π = 360◦ , then π = 180◦
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Radians and Degrees
What relationships exist between degrees and radians?
If 2π = 360◦ , then π = 180◦
Using this we get the following conversion factors:
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Radians and Degrees
What relationships exist between degrees and radians?
If 2π = 360◦ , then π = 180◦
Using this we get the following conversion factors:
π
degrees to radians: multiply by
180
radians to degrees: multiply by
180
π
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Radians and Degrees
Example: Convert 225◦ into radians.
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Radians and Degrees
Example: Convert −
7π
into degrees.
4
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Coterminal Angle
Two angles in standard position are said to be coterminal if they share the
same initial and terminal sides.
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Coterminal Angle
Two angles in standard position are said to be coterminal if they share the
same initial and terminal sides.
The angles θ =
π
7π
and γ =
are coterminal.
3
3
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Coterminal Angle
Two angles in standard position are said to be coterminal if they share the
same initial and terminal sides.
The angles θ =
π
7π
and γ =
are coterminal.
3
3
Note: Angles are coterminal if they differ by a multiple of 360◦ or 2π
radians.
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Coterminal Angles
Example: Find an angle that is coterminal to 1230◦ .
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Coterminal Angles
π
Example: Find an angle that is coterminal to − .
8
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Arc Length
Given a circle of radius r, the length of the arc intercepted by a central
angle θ is given by
s = rθ
s
θ
r
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Arc Length
Example: Find the length of the arc made by a 220◦ angle on a circle with
radius 9 in.
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Arc Length
Example: Lincoln, Nebraska, is located at 40.8◦ N, 96.7◦ W and Dallas,
Texas, is located at 32.8◦ N, 96.7◦ W. Using the difference in latitudes,
approximate the distance between the cities assuming that the radius of
the Earth is 3960 mi. Round to the nearest mile.
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Linear and Angular Speed
If we consider the spokes of a bicycle tire, we have two ways to measure
the speed. The first is, we can consider the linear speed, or how fast a
point on the tire is moving. We can also measure the angular speed, or the
rate at which the angle of the tire changes.
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Linear and Angular Speed
If we consider the spokes of a bicycle tire, we have two ways to measure
the speed. The first is, we can consider the linear speed, or how fast a
point on the tire is moving. We can also measure the angular speed, or the
rate at which the angle of the tire changes.
If a point on a circle of radius r moves through an angle of θ radians in
time t, the angular and linear speeds of the point are
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Linear and Angular Speed
If we consider the spokes of a bicycle tire, we have two ways to measure
the speed. The first is, we can consider the linear speed, or how fast a
point on the tire is moving. We can also measure the angular speed, or the
rate at which the angle of the tire changes.
If a point on a circle of radius r moves through an angle of θ radians in
time t, the angular and linear speeds of the point are
ω=
θ
t
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Linear and Angular Speed
If we consider the spokes of a bicycle tire, we have two ways to measure
the speed. The first is, we can consider the linear speed, or how fast a
point on the tire is moving. We can also measure the angular speed, or the
rate at which the angle of the tire changes.
If a point on a circle of radius r moves through an angle of θ radians in
time t, the angular and linear speeds of the point are
θ
t
s
rθ
v= =
= r ω.
t
t
ω=
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Linear and Angular Speed
Example: A bicycle wheel rotates at 2 revolutions per second. Find the
angular speed. If the wheel is 2.2 feet in diameter, what is the speed of
the bike in miles per hour?
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Area of a Sector
In the following image, can we find the area of the region enclosed by the
angle inside the circle?
s
θ
r
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Area of a Sector
Example: A sprinkler rotates through an angle of 120◦ and sprays water a
distance of 30 ft. Find the amount of area watered. Round to the nearest
whole unit.
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