Embed Size (px)
Citation preview
Chapter 4 Circular and
Trigonometric Functions
1
Chapter 4.1 The Unit Circle
2
Unit Circle
2 2
A is a circle whose radius is equal to
one unit and whose center is at the origin.
Every point on the unit circle satisfies the
equation
unit circle
1.x y
3
-1.5 -1.0 -0.5 0.5 1.0 1.5
-1.5
-1.0
-0.5
0.5
1.0
1.5
x
y
1,0
0,1
1,0
0, 1
4
-1.5 -1.0 -0.5 0.5 1.0 1.5
-1.5
-1.0
-0.5
0.5
1.0
1.5
x
y
Example 4.1.1
2 2
2
2
2
2
1Find points in the unit circle whose coordinate is .
2
1 The points are
1 1 3 1 31 , & , .
2 2 2 2 2
11
43
4
3
2
x
x y
y
y
y
y
1 3,
2 2
1 3,
2 2
5
Arcs
are parts of a circle with two endpoints,
the initial point and terminal po
Arcs
int.
initial point
terminal point
6
Angles
The is formed by two rays
and . The poin
plane angle
vert is called the
and the rays are called
tex
sidthe .es
NOP
ON OP O
N
O P
7
O Pinitial side
N
terminal side
An angle is if the direction of rotation
is counterclockwise
and if the direction of rotati
positive
n on is
cloc
egat
kw
ive
ise. 8
The intercepts the .
The subtends
arc
ar the
angle
ac ngle.9
Degrees
A is defined as the measure of a
central angle subtended by an arc of a circle
1equal to of the circumference of the36
degree
0
circle.
10
1 of the 360
circumference
The measure of the central angle is 1
11
Radians
A rad is defined as the measure of the
central angle subtended by an arc of a circle
equal to the radius of the ci
radian
rcle.
12
arc whose
length is r
The measure of the central angle is 1 rad.
r
13
Degrees and Radians
The circumference of a circle is 2 and it
subtends an angle of 360 .
2 r
1801 rad
1 r
ad 3
a
0
d180
6
r
14
Example 4.1.2
71. Express rad in degrees.
12
7 7 180 rad 105
12 12
2. Express 210 in radians.
7210 210 rad rad
180 6
1801 rad
1 rad180
15
Arc Length
On a circle of radius , a central angle of
intercepts an arc of length
.r
r
s
r
s r
16
Arc Length in a Unit Circle
In a unit circle, 1.r
s
1
17
End of Chapter 4.1
18
