6.6 The Unit Circle
OBJECTIVE 1: Understanding the Definition of the Unit Circle
The Unit Circle
A circle centered at the origin with a radius of 1 unit is called the unit circle
whose equation is given by x 2  y 2  1. .
OBJECTIVE 2: Using Symmetry to Determine Points that Lie on the Unit Circle
OBJECTIVE 3: Understanding the Unit Circle Definitions of the Trigonometric Functions
So far in this text, we have seen two groups of definitions for the trigonometric functions. In Section 6.4 we
saw the Right Triangle Definitions of the Trigonometric functions of acute angles . Then, in Section 6.5 we
saw the Definitions of the Trigonometric Functions of General Angles.
We now turn our attention to the third definition of trigonometric functions that involve the unit circle. Suppose
that t is the measure (in radians) of a central angle of a unit circle with a corresponding arc length, s.
s
We can use the formula for the arc length of a sector of a circle to find the arc length, s of a unit circle that
corresponds to a central angle of t radians.
s  r
s  1 t
s t
Write the formula for the arc length of a sector of a circle.
Substitute r  1 and   t .
Simplify.
We see that the length of the intercepted arc is s  t . This means that the arc length of a sector of the unit
circle is exactly equal to the measure of the central angle! The arc length and the angle are represented by
the same real number t.
We can now define the unit circle definitions of the trigonometric functions. To do this, let t be any real
number and let P ( x, y ) be the point on the unit circle that has an arc length of t units from the point (1, 0) . The
measure of the central angle (in radians) is exactly the same as the arc length, t. If t  0 , then point P is
obtained by rotating in a counterclockwise direction. If t  0 , then point P is obtained by rotating in a
clockwise direction.
For the real number t and the corresponding point P ( x, y ) lying on the graph of the unit circle, we define the
cosine of t as the x-coordinate of P and the sine of t as the y-coordinate of P. Therefore, cos t  x and sin t  y .
This choice of x for the cosine of t and y for the sine of t is not made arbitrarily. If you remember the
y
Definitions of the Trigonometric Functions of General Angles, the sine of an angle theta was defined as
and
r
x
the cosine of theta was defined as . Therefore, it seems logical and consistent that we choose sin t to be y
r
y
x
(which is
when r  1 ) and cos t to be x (which is
when r  1 ).
r
r
We now define all six of the trigonometric functions using the unit circle.
The Unit Circle Definitions of the Trigonometric Functions
For any real number t, if P ( x, y ) is a point on the unit circle
corresponding to t, then
sin t  y
cos t  x
tan t 
y
, x0
x
1
, y0
y
1
sec t  , x  0
x
x
cot t  , y  0
y
csc t 
OBJECTIVE 4: Evaluating Trigonometric Functions for Increments of

2
If we traverse the circle one complete revolution, an arc length of 2 , we are back at the point, 1,0  . Thus,
5
7
 1 , sin 3  0 , sin
 1 , and
2
2
sin 4  0 . We can continue traversing the unit circle indefinitely. This is why the trigonometric functions are
often referred to as “circular functions.”
sin 2  0 . We can continue around the circle again to obtain sin
Because division by zero is undefined, the values of trigonometric functions may be undefined. For example,

 y 1
when t  , the corresponding point on the unit circle is (0,1) . Thus, tan   , which is undefined.
2
2 x 0
 3
Table 3 shows the values of the six trigonometric functions for values of 0 , ,
, and 2 .
2 2
OBJECTIVE 5: Evaluating Trigonometric Functions at Increments of
  
,
,
6 4
, and

3
  
, , right
4 4 2
3 6 2
triangle. We can use our knowledge of these two special right triangles to determine several more points that lie
on the unit circle.
In Section 6.3 we introduced two special right triangles, the
,
 
right triangle and the
 1 1 
If we superimpose this triangle onto the unit circle, we see that the point 
,
 lies on the unit circle.
 2 2
1 3
  
Similarly, we can use our knowledge of the , , right triangle to show that the points  ,
 and
3 6 2
2 2 
 3 1
,  also lie on the unit circle.

 2 2
 3 1
 1 1  1 3
,
,  lie on the unit circle, we can use the
Now that we know that the points 
,
and
,




 2 2   2 2 
 2 2
symmetry properties of the unit circle to complete the unit circle . This unit circle contains the “special points”
that correspond with “special values of t.”