Definition 3:
Trigonometric Functions:
The Unit Circle
3.4
JMerrill, 2009
Contributions from DDillon
Recall – Definitions of Trig
Functions


Definition 1 involved the ratios of 2 sides
of a triangle (SOH CAH TOA)
Definition 2 dealt with ratios using x- and
y-coordinates and the distance from the
origin to a point (using x’s, y’s, and r’s)
The Unit Circle
This circle has radius of 1.
(0,1)
It is centered at the origin.
Endpoints are labeled as
(1, 0)
(-1, 0)
This is the standard that
we use. All our function
values are based on this
standard.
(0, -1)
Definition 3: The Unit Circle

Let (x, y) be any point on the unit circle. If θ is the
central angle that has the same measure as the arc
length from the point (1,0) along the circumference to
the point (x, y), then
y y
 y
r 1
x
cot   y  0
y
sin  

x x
 x
r 1
1
sec   x  0
x
cos  
y
x0
x
1
csc   y  0
y
tan  
The coordinates of the points along the unit circle
can be written (cosθ, sinθ).
Trig Function Values of
Quadrantal Angles
x’s are cosines
(0,1)
y’s are sines
0
1. sin 180º = _____
(1, 0)
(-1, 0)
0
2. cos 90º = _____
0
3. cot 270º = _____
undef
4. tan 90º = _____
(0, -1)

1
5. csc = _____
2
1
6. Sec 2 = _____
Recall: 45°-45°-90° Triangles
In any 45°-45°-90° triangle, the sides are in the ratio 1 :1 : √2.
sin 45° = √2/2
cos 45° = √2/2
45°
1
tan 45° = 1
√2/2
45º
√2/2
The Unit Circle
Recall: 30°-60°-90° Triangles
In any 30°-60°-90° triangle, the sides are in the ratio 1 : 2 : √3
sin 60° = √3/2
cos 60° = 1/2
30°
1
tan 60° = √3
√3/2
sin 30° = 1/2
cos 30° = √3/2
60º
tan 30° = √3/3
1/2
The Unit Circle
Trig Function Values
1/2
1. sin 30º = _______
4. cos π/4 =
√2/2
_______
√3
2. tan π/3 = _______
5. sec π/6 =
2√3/3
_______
0
_______
6. cot π/2 =
0
_______
3. sin π =