4.2 Trigonometric Function:
The Unit circle
The Unit Circle
A circle with radius of 1
Equation x2 + y2 = 1
0,1
 1,0

0,1
cos , sin  
1,0
The Unit Circle with Radian Measures
2
Do you remember 30º, 60º, 90º
triangles?
Now they are really! Important
Do you remember 30º, 60º, 90º
triangles?
Now they are really! Important
Even more important
Let 2a = 1
Do you remember 30º, 60º, 90º
triangles?
Let 2a = 1
3
Cos30 
2
1
Sin 30 
2
Do you remember 30º, 60º, 90º
triangles?
1
Cos60 
2
3
Sin 60 
2
1
3

1
2
Do you remember 45º, 45º, 90º
triangles?
When the hypotenuse is 1
The legs are 2
2
2
Cos45 
2
2
Sin 45 
2
1
2
2
2
2
Some common radian
measurements
These are the Degree expressed in Radians
30 
45 
60 

6

4

3
90 

2
180  
3
270 
2
360  2
The Unit Circle: Radian Measures and Coordinates
2
The Six Trig functions
b
adjacent

c hypotenuse
a
opposite
Sin   
c hypotenuse
a opposite
Tan  
b adjacent
Cos 
1
Cos 
 Sec
cos
1
Sin 
 Csc
sin
1
Tan 
 Cot
tan
Sin 
Tan 
Cos

Why does the book use “t” for an
angle?
Since Radian measurement are lengths of
an arc of the unit circle, it is written as if
the angle was on a number line.
Where the distance is “t’ from zero.
Later when we graph Trig functions it just
works better.
Lets find the Trig functions if
2

3
Think where this angle is on the unit circle.
 2
Cos
 3
 1 3 
 ,

 2 2 


2

3
 1

 2
3
 2 
Sin 

 3  2
3
 2 
Tan
 2  3
 3  1
2
Sin 
Tan 
Cos
Find the Trig functions of
2

3
Think where this angle is on the unit circle.
 2
Cos
 3
 1

 2
 2   2
 2
Sec

 3  1
 2
Sin 
 3
3



 2
2 3
 2  2

Csc

3
3
 3 
3
 2 
Tan
 2  3
 3  1
2
 2   1  3

Cot 

3
3
 3 
How about


4
 2 2


 2 , 2 




4
 2  2


 2 , 2 


 
Cos
 4
2



 2
 
Sin
 4
  2

2

2
  
Tan
  2  1
 4   2
2
There are times when Tan or Cot
does not exist.
At what angles would this happen?
There are times when Tan or Cot does not exist.
 3 
 , 
2 2 
If think of the domain of the trig
functions, there are some limits.
Look at the unit circle. If x goes with Cos,
then what are the possible of Cos?
It is the same with
Sin?
Definition of a Periodic Function
A function “f” is periodic if there exist a
positive real number “ c” such that
f(t + c) = f(t) for all values of “t”.
The smallest “c” is called the period.
Even Function ( Trig. )
Cos (- t) = Cos (t) and Sec( -t) = Sec (t)
Also
Sin(-t) = -sin (t) and Csc (-t) = - Csc (t)
Tan(-t) = -Tan (t) and Cot(-t) = - Cot (t)
Homework
Page 278- 279
#1, 5, 9, 13, 17,
21, 25, 29, 33,
37, 41, 45, 48,
52, 59, 68
Homework
Page 278- 279
# 2, 8, 12, 16,
20, 24, 28, 32,
36, 40, 44, 49,
58, 61