©
:
Pre

Calculus

Chapter 8A
Chapter 8A

Angles and Circles
Many applications of calculus use trigonometry, which is the study of angles and functions of angles
and their application to circles, polygons, and science. We start with the definition of angles and their
measures.
Angles
Roughly, an
angle
is the space between two rays or line segments with a common endpoint. The rays
or line segments are called the sides and the common endpoint is called the vertex. More precisely, if a
ray or line segment rotates about an endpoint from some initial position, called the initial side, to some
final position, called the final side, then the angle between the sides is the space swept out. Within a
plane, we say the angle is positive if the rotation is counterclockwise, and is negative if the rotation is
clockwise. An angle is in standard position if the vertex is at the origin and the initial side is along the
positive
x
axis.
θ
A positive angle
θ
x
y
An angle in standard
θ
A negative angle
Measures of Angle
The size of an angle may be measured in revolutions (rev), in degrees (°) or in radians (rad).
An angle is called a full rotation if the ray rotates from the initial side all the way around so that the
final side coincides with the initial side.
A full rotation is measured as 1rev
360°
2
rad.
An angle is called a half rotation or a straight angle if the ray rotates from the initial side to a final side
which is directly opposite to the initial side.
A half rotation is measured as
1
2
rev
180°
rad.
An angle is called a quarter rotation or a right angle if the ray rotates from the initial side to a final side
which is perpendicular to the initial side.
A quarter rotation is measured as
1
4
rev
90°
2
rad.
An angle is called a null rotation if the ray never rotates so that the final side coincides with the initial
side.
A null rotation is measured as 0rev
0°
0rad.
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Calculus

Chapter 8A
Several other important angles are:
3
4
rev
270°
3
2
rad
1
6
rev
60°
3
rad
1
8
rev
45°
4
rad
1
12
rev
30°
6
rad
Example 1
:
Find the angular measure of one time zone on the surface of the earth.
Solution:
The earth rotates once a day or by 1rev in 24 hours. Thus a 1 hour time zone has an
angular measure of
1
24
rev
15°
12
rad.
An angle will be bigger than a full rotation if the ray rotates from the initial side all the way around and
past the initial side again. There is no limit to the size of an angle either positive or negative. Two
angles with the same initial and final sides are called
coterminal
and their measures must differ by an
integral multiple of 1rev
360°
2
rad.
Below are some pictures of various angles. In each figure, the initial sides of the angles coincide.
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 Fall '07
 GENERALCHEMFORENG
 Trigonometry, Pythagorean Theorem, Sin, Cos

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