Chapter 5 – Trigonometric Functions:
Unit Circle Approach
5.1 - The Unit Circle
The Unit Circle

The unit circle is the circle of radius 1 centered at the
origin in the xy-plane. Its equation is
x  y 1
2
2
5.1 - The Unit Circle
Example – pg. 375

Show that the point is on the unit circle.
 5 12 
4.   , 
 13 13 
 5 2 6
6.   ,

 7 7 
 11 5 
8. 
, 
 6 6
5.1 - The Unit Circle
Terminal Points on the Unit
Circle

Suppose t is a real number. Let t be the distance
along the unit circle starting at the point (1, 0) and
ending at the point P (x, y). This point is the
terminal point determined by the real number t.
5.1 - The Unit Circle
Terminal Points

The circumference of a circle is C = 2. So if a point
starts at (1, 0) and moves counterclockwise all the
way around the circle, it travels a distance of t = 2.
5.1 - The Unit Circle
Terminal Points

The circumference of a circle is C = 2. So if a point
starts at (1, 0) and moves counterclockwise half of
the way around the circle, it travels a distance of
1
t   2   
2
5.1 - The Unit Circle
Terminal Points

The circumference of a circle is C = 2. So if a point
starts at (1, 0) and moves counterclockwise a quarter
of the way around the circle, it travels a distance of
1

t   2  
4
2
5.1 - The Unit Circle
Terminal Points
The circumference of a circle is C = 2. So if a point
starts at (1, 0) and moves counterclockwise three
quarters of the way around the circle, it travels a
distance of
3
3
t   2  
4
2

5.1 - The Unit Circle
Terminal Points
5.1 - The Unit Circle
Examples – pg. 376

Find the terminal point P (x, y) on the unit circle
determined by the given value of t.
3
24. t 
2
7
26. t 
6
27. t  

3
5.1 - The Unit Circle
Example – pg. 376

Suppose that the terminal point determined by t is the
point
on the unit circle.

Find the terminal point determined by each of the
following.
(a) -t
(b)  - t
(c) 4 + t
(d) t - 
5.1 - The Unit Circle
Reference Number

Let t be a real number. The reference number`t
associated with t is the shortest distance along the
unit circle between the terminal point determined by t
and the x-axis.
5.1 - The Unit Circle
Examples – pg. 376

Find the reference number for each value of t.
5.1 - The Unit Circle
Using Reference Numbers to
Find Terminal Points

To find the terminal point P by any value of t, we use
the following steps:
1.
Find the reference number`t.
2.
Find the terminal point Q (a, b) determined by`t.
3.
The terminal point determined by t is P (±a, ±b),
where the signs are chosen according to the
quadrant in which this terminal point lies.
5.1 - The Unit Circle
Examples – pg. 376

Find (a) the reference number for each value of t and
(b) the terminal point determined by t.
7
42. t 
3
7
44. t  
6
13
46. t 
6
17
48. t 
4
31
50. t 
6
41
52. t  
4
5.1 - The Unit Circle