Download 6.5 Circular Functions

CHAPTER 6:
The Trigonometric Functions
6.1
6.2
6.3
6.4
6.5
6.6
The Trigonometric Functions of Acute Angles
Applications of Right Triangles
Trigonometric Functions of Any Angle
Radians, Arc Length, and Angular Speed
Circular Functions: Graphs and Properties
Graphs of Transformed Sine and Cosine
Functions
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6.5
Circular Functions: Graphs and Properties




Given the coordinates of a point on the unit circle, find
its reflections across the x-axis, the y-axis, and the
origin.
Determine the six trigonometric function values for a
real number when the coordinates of the point on the
unit circle determined by that real number are given.
Find the function values for any real number using a
calculator.
Graph the six circular functions and state their
properties.
Copyright © 2009 Pearson Education, Inc.
Unit Circle
We defined radian measure to be
s

r
When r = 1,
s
  , or   s
1
The arc length s on a unit circle is the same as the
radian measure of the angle .
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Slide 6.5 - 4
Basic Circular Functions
For a real number s that determines a point (x, y) on the
unit circle:
sin s  second coordinate  y cos s  first coordinate  x
second coordinate
tan s 

first coordinate
1
csc s 

second coordinate
y
x
1
y
1
1
sec s 

first coordinate x
x  0 
first coordinate
x
cot s 

second coordinate y
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x  0 
y  0 
y  0 
Slide 6.5 - 5
Reflections on a Unit Circle
Let’s consider the radian measure π/3 and determine
the coordinates of the point on the unit circle.
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Slide 6.5 - 6
Reflections on a Unit Circle
We have a 30º- 60º right triangle with hypotenuse 1
and side opposite 30º 1/2 the hypotenuse, or 1/2. This
is the x-coordinate of the point. Let’s find the
y-coordinate.
2
 1
2

y
1
 
2
1 3
y  1 
4 4
2
y
3
3

4
2
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Slide 6.5 - 7
Example
Each of the following points lies on the unit circle. Find
their reflections across the x-axis, the y-axis, and the
origin.
 2 2
 1 3
 3 4
a)  , 
b) 
,
c)  ,

 5 5
 2 2 
 2 2 
Solution:
a)
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Slide 6.5 - 8
Example
Solution continued
 2 2
b) 
,
 2 2 
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Slide 6.5 - 9
Example
Solution continued
 1 3
c)  ,
 2 2 
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Slide 6.5 - 10
Find Function Values
Knowing only
a few points
on the unit
circle allows
us to find
trigonometric
function
values of
frequently
used numbers.
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Slide 6.5 - 11
Example
Find each of the following function values.
a) tan

3
 
c) sin   
 6
e) cot 
3
b) cos
4
4
d) cos
3
 7 
f) csc   
 2 
Solution
Locate the point on the unit circle determined by the
rotation, and then find its coordinates using reflection
if necessary.
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Slide 6.5 - 12
Example
Solution continued
a) tan

3

y
tan 
3 x
3 2

12
 3
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Slide 6.5 - 13
Example
Solution continued
3
b) cos
4
3
cos
x
4
2

2
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Slide 6.5 - 14
Example
Solution continued
 
c) sin   
 6
 
sin     y
 6
1

2
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Slide 6.5 - 15
Example
Solution continued
 4 
d) cos 
 3 
 4 
cos 
x

 3 
1

2
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Slide 6.5 - 16
Example
Solution continued
e) cot 
x
cot  
y
1

0
which is not defined
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Slide 6.5 - 17
Example
Solution continued
 7 
f) csc   
 2 
 7  1
csc    
 2  y
1

1
1
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Slide 6.5 - 18
Example
Find each of the following function values of radian
measures using a calculator. Round the answers to four
decimal places.

2
d) sec
c) sin 24.9
b) tan 3
a) cos
7
5
Solution:
With the calculator in RADIAN mode:
2
a) cos
 0.3090
5
b) tan 3  0.1425
c) sin 24.9  0.2306
d) sec
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
7

1
cos

7
 1.1099
Slide 6.5 - 19
Graph of Sine Function
Make a
table of
values
from the
unit
circle.
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Slide 6.5 - 20
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Slide 6.5 - 21
Graph of Sine Function
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Slide 6.5 - 22
Graph of Cosine Function
Make a
table of
values
from the
unit
circle.
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Slide 6.5 - 23
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Slide 6.5 - 24
Graph of Cosine Function
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Slide 6.5 - 25
Domain and Range of Sine and Cosine
Functions
The domain of the sine function and the cosine
function is (–∞, ∞).
The range of the sine function and the cosine function
is [–1, 1].
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Slide 6.5 - 26
Periodic Function
A function with a repeating pattern is called periodic.
The sine and cosine functions are periodic because
they repeat themselves every 2π units.
To see this another way, think of the part of the graph
between 0 and 2π and note that the rest of the graph
consists of copies of it.
The sine and cosine functions each have a period of 2π.
The period can be thought of as the length of the
shortest recurring interval.
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Slide 6.5 - 27
Periodic Function
A function f is said to be periodic if there exists a
positive constant p such that
f s  p   f s 
for all s in the domain of f. The smallest such positive
number p is called the period of the function.
sin s  2   sin s
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cos s  2   cos s
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Amplitude
The amplitude of a periodic function is defined to be
one half the distance between its maximum and
minimum function values. It is always positive.
Both the graphs and the unit circle verify that the
maximum value of the sine and cosine functions is 1,
whereas the minimum value of each is –1.
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Slide 6.5 - 29
Amplitude of the Sine Function
1
the amplitude of the sine function  1  1  1
2
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Slide 6.5 - 30
Amplitude of the Cosine Function
1
the amplitude of the cosine function  1  1  1
2
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Slide 6.5 - 31
Odd and Even
Consider any real number s and its opposite, –s. These
numbers determine points T and T1.
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Slide 6.5 - 32
Odd and Even
Because their second coordinates are opposites of each
other, we know that for any number s,
sin s    sin s
Because their first coordinates are opposites of each
other, we know that for any number s,
cos s   cos s
The sine function is odd.
The cosine function is even.
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Slide 6.5 - 33
Graph of the Tangent Function
Instead of a table, let’s begin with the definition and a
few points on the unit circle.
y sin s
tan s  
x cos s
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Slide 6.5 - 34
Graph of the Tangent Function
Tangent function is not defined when x, the first
coordinate, is 0; that is, when cos s = 0:

3
5
s ,
,
, ...
2
2
2
Draw vertical
asymptotes at these
locations.
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Slide 6.5 - 35
Graph of the Tangent Function
Note:
tan s  0 at s  0,   ,  2 ,  3 ,...
7
3  5 9
tan s  1 at s  ... 
,
, ,
, ...
4
4 4 4 4
9
5
 3 7
tan s  1 at s  ... 
,
, ,
, ...
4
4
4 4 4
Add these ordered
pairs to the graph.
Use a calculator to
add some other
points in (–π/2, π/2).
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Slide 6.5 - 36
Graph of the Tangent Function
Now we can complete the graph.
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Slide 6.5 - 37
Graph of the Tangent Function
From the graph, we see that:
Period is π.
There is no amplitude (no maximum or minimum
values).
Domain is the set of all real numbers except (π/2) + kπ,
where k is an integer.
Range is the set of all real numbers.
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Slide 6.5 - 38
Graph of the Cotangent Function
The cotangent function (cot s = cos s/sin s) is not
defined when y, the second coordinate, is 0; that is, it is
not defined for any number s whose sine is 0.
Cotangent is not defined for s = 0, ±2π, ±3π, …
The graph of the cotangent function is on the next
slide.
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Slide 6.5 - 39
Graph of the Cotangent Function
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Slide 6.5 - 40
Graph of the Cotangent Function
From the graph, we see that:
Period is π.
There is no amplitude (no maximum or minimum
values).
Domain is the set of all real numbers except kπ, where
k is an integer.
Range is the set of all real numbers.
Copyright © 2009 Pearson Education, Inc.
Slide 6.5 - 41
Graph of the Cosecant Function
The cosecant and sine functions are reciprocals.
The graph of the cosecant function can be constructed
by finding the reciprocals of the values of the sine
function. The cosecant function is not defined for those
values of s whose sine is 0.
The graph of the cosecant function is on the next slide
with the graph of the sine function in gray for
reference.
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Slide 6.5 - 42
Graph of the Cosecant Function
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Slide 6.5 - 43
Graph of the Cosecant Function
From the graph, we see that:
Period is 2π.
There is no amplitude (no maximum or minimum
values).
Domain is the set of all real numbers except kπ, where
k is an integer.
Range is (–∞, –1] U [1, ∞).
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Slide 6.5 - 44
Graph of the Secant Function
The secant and cosine functions are reciprocals.
The graph of the secant function can be constructed by
finding the reciprocals of the values of the cosine
function. The secant function is not defined for those
values of s whose cosine is 0.
The graph of the secant function is on the next slide
with the graph of the cosine function in gray for
reference.
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Slide 6.5 - 45
Graph of the Cosecant Function
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Slide 6.5 - 46
Graph of the Secant Function
From the graph, we see that:
Period is 2π.
There is no amplitude (no maximum or minimum
values).
Domain is the set of all real numbers except kπ, where
k is an integer.
Range is (–∞, –1] U [1, ∞).
Copyright © 2009 Pearson Education, Inc.
Slide 6.5 - 47