Download Lecture 1 (L1): Angles and Angle Measures Textbook Section: 4.1

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Lecture 1 (L1): Angles and Angle Measures
Textbook Section: 4.1
Geometry Review:
Point
Line segments
Ray
Vertex
Cartesian plane
Angles:
An __________________________ is formed by rotating a _______________ about its endpoint or
________________________. The starting position is called the ____________________________ of the
angle and the ending position is called the ______________________________.
Angles in the Coordinate Plane:
An angle is said to be in __________________________________ if its initial side is located on
the ________________________________________ , its vertex at the _______________________ , and has been
rotated _______________________________ about its vertex. _________________________ angles have the
same initial side and vertex position, but have been rotated____________________________________.
Notation and examples:
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Angle Measure:
Angles are measured by the amount of _____________________ from initial to terminal
side. When measuring angles on the coordinate plane it is convenient to imagine the angle
sitting inside of a _______________________ because if you rotate an angle in the coordinate plane
completely around its vertex, the path the angle makes forms a _____________________.
A _________________________ is defined as _____________ of a complete rotation. A complete
rotation measures_______________________, a half rotation measures______________________, a
quarter rotation measures__________________________________, and three-quarters rotation
measures____________________________.
Diagram:
Another way of measuring angles is using ____________________________.
One___________________ is the measure of a central angle 𝜃 that intercepts
__________________________________that has length equal to _______________________________________.
Diagram and algebraic representation:
Problem: How many radians is a full revolution? How many radians is a half, a quarter, and
three-quarters revolution?
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Problem: How do we convert between different angle measures?
Angle pairs:
Two positive angles are said to be ___________________________________ if their sum is
________________ or ________________. Two angles positive are said to be ___________________________
if their sum is _________________ or _____________________.
Problem: Find the supplement and complement of
2𝜋
3
.
Angles that have the same ______________________ and ________________________ sides are
said to be _____________________________________.
Problem: Find two angles, one positive and one negative, that are conterminal with
3
2𝜋
3
.
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Arc Length:
Recall that the radian measure of an angle 𝜃 is given by ________________________________
where _____ represents the ________________________________________________________ and ______ is the
_________________________________________________.
Problem: How can we use the formula for radian measure to find a formula for the arc
length of a circle?
Problem: Find the length of the arc of a circle with radius 3 intercepted by a central angle
of 140° .
NYTI Problem: Given a circle with radius 20, find the length of the arc intercepted by a
central angle of 250° .
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Problem: Say a particle is moving at a constant speed along a circular arc of radius 2. If the
particle travels an arc length of 4 units in 3 seconds, how fast is the particle moving along
the circle?
Linear and Angular speed:
The ____________________________________ of a particle moving at a constant rate along a
circular arc with a radius of _______ is the ____________________________________________ traveled in
_________________ .
Diagram and algebraic representation:
The __________________________________________ of the particle is found by considering the
measure of _______________________ corresponding to the arc length 𝑠 traveled in time 𝑡.
Diagram and algebraic representation:
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Problem: If the minute hand of a clock is 3 inches long, what is the linear and angular
speed of the minute hand?
Problem: Two pulleys are connected by a moving belt. One pulley has radius 3 cm and the
other has radius 9 cm. If the smaller pulley is rotating at 6 revolutions per minute, how
many revolutions per minute is the larger pulley rotating?
A _________________________ of a circle is a part of the circle that is bounded by
____________________________________________________ and their _______________________________________.
Problem: How can we find the area of a sector of a circle given the formula for the area of a
circle with radius 𝑟 is 𝐴 = 𝜋𝑟 2 ?
Area of a sector of a circle:
The area of a sector of a circle 𝐴 with radius 𝑟 and central angle 𝜃, in
__________________, is given by ________________________________________________.
Problem: On a clock with a minute hand 3 inches long, how much area of the clock face is
covered by the minute hand in 15 minutes?
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