Conditional and
Absolute Value
Functions
Objectives
Conditional
Determine
Sketch
its domain and range
the graph
Absolute
value functions
Determine
Sketch
functions
its domain and range
the graph
Definition
A conditional (or piecewise)
function is a fu
Double Measure
Identities
Double Measure
Identities
Double Measure
Identities
If
, then
Double Measure
Identities
If
, then
sin sin cos cos sin
sin 2 sin cos sin cos
2sin cos
tan tan
tan
1 tan tan
2 tan
tan 2
2
1 tan
Example
Given:
Determine
S
GRAPHS OF
TANGENT AND
COTANGENT FUNCTIONS
yy
yy
O
x
If a>0
O
x
If a<0
EXAMPLE
k
Dom f R
k is odd integer
2
0
4
2
3
4
EXAMPLE
k k
Dom f R
k is odd integer
2 2
0
1
4
1
2
3
4
1
TRY THESE
yy
yy
O
x
If a>0
O
x
If a<0
EXAMPLE
EXAMPLE
k
Dom f R
| k Z
Identities
Sum and Difference
Identities
Illustration 1:
Determine the exact value of
7
sin
12
7
cos
12
7
tan
12
Illustration:
Solution:
7
sin
12
sin
3 4
sin cos cos sin
3
4
3
4
3 2 1 2
2 2 2 2
6
2
6 2
4
4
4
Illustration
7
cos
12
Circular
Functions
Recall
The wrapping function P is a
function from
to U such that
where (x, y) is the terminal
point of the arc with length t.
Is P one-to-one?
Special Real Numbers
Examples
Examples
Sine Function
Definition
Let P be the wrapping functio
Inverse
Circular
Functions
Restricting the Domain
In order to define the inverse circular
functions, we will have to restrict the
domains of the circular functions.
This is to make the circular functions
one-to-one in the restricted domains.
Inverse Sine
Circular Functions
Are circular
functions one-toone?
Sine Function
Cosine Function
Periodic
Circular functions are periodic.
Their function values repeat
after a fixed interval.
The length of fixed interval is
called the period, denoted by
p.
A cycle is s
Establishing
Identities
Establishing Identities
Start with one side of the
given identity.
We usually choose the more
complicated side.
Apply the appropriate
identities to derive the other
side.
Example
Prove:
Proof:
Example
Prove:
Example
Prove:
Proof:
Identities
Recall
Fundamental Identities
Fundamental Identities
Points on the unit circle satisfy
the equation
Then we have
Dividing both sides of the
equations with
, we get
Fundamental Identities
Dividing by
Pythagorean
Identities
, we get
Example
Find
Circular Functions
Unit Circle
The unit circle is the circle
centered at the origin and has
radius 1.
The Unit Circle
(0,1)
(-1,0)
(1,0)
(0,-1)
Arcs
Arcs are parts of a circle
with two endpoints, the initial
point and terminal point.
Wrapping Function
We