Transcript Sec 2.1

Chapter 2
Trigonometric Functions of Real
Numbers
Section 2.1
The Unit Circle
y
The Unit Circle
x2  y 2  1
The unit circle is a circle of radius 1 with it center at
the origin. The equation of the unit circle is x2+y2=1.
Any point on the unit circle will have the sum of the
squares of its x and y coordinates equal to 1.
x
1
Terminal Points and Radian Measure
A terminal point on the unit circle is a point on
the unit circle that forms an angle with the
positive x-axis. The distance you travel on the
unit circle starting from the point (1,0) on the
positive x-axis and ending at the terminal point
(x0,y0) is the radian measure of the angle.
(Remember the measure is positive if you move
counterclockwise and negative if you move
clockwise.) The radian angle measure we
usually denote with the letter t.
 t
x0 , y0 

1
The length of the red arc
above is the radian measure
of the angle in standard
position with the point (x0,y0)
on it terminal side.
t

2
t 
1
1
t
3
2
t

t
4
1
1
1

t
2
3
4
1
 5
t
4
1
The pictures above illustrate different angles on the unit circle along with their radian
measure which is also the length of the red arc. What are the measures of the last 4
angles?
Points and Angles
t
There are some angles on the unit circle for
which we know the coordinates of the terminal
point. These come from realizing the triangle
formed by the terminal point the point
perpendicular on the x-axis and the origin is
either a 30°-60°-90° triangle or a 45°-45°-90°
triangle. In radians we would say they are:
  
/
/
6 3 2
or
/
4 4 2
2
; (0,1)
t
1 3

;  ,
3  2 2 

t
 2 2

; 
,
4  2 2 

t
  
/

 3 1
; 
, 
6  2 2 

t  0; (1,0)
0
1
Reference Number
A reference number is similar to a reference angle. The reference number for a
given number is the shortest distance you would need to travel to get to the x-axis
on either the positive or negative side. The reference numbers can be used to
calculate the coordinates of various points on the unit circle. The reference number
for t is usually denoted
t
y
t

3
P
t
y
2
3
t
5
4
x
x
t

4
P
The reference number for: t 
is the number:
t

3
The point P has
coordinates given by:
2
3
The reference number for: t 
is the number:
 1 3 
 ,

 2 2 


t
5
4

4
The point P has
coordinates given by:
 2  2


 2 , 2 


Example 1
Example 2
Find the point on the unit circle whose
radian measure is 5 .
A point in the third quadrant has an
x-coordinate of -⅓. Find its ycoordinate.
3
I begin by drawing a circle.
The reference angle is:

3
The triangle that is made is :
The coordinates of P are:
y
  
/
P :  , y
/
6 3 2
1  3
 ,

2 2 


Take the negative
root because y is
negative in the
third quadrant.
1
9
 y2  1
y2 
y
y
5
t
3
x
1
3

t
3
P
 31 2  y 2  1
1
3
P
x
8
9
 8
3