Lesson 4.2
A circle with center at (0, 0) and radius 1 is called a unit circle.
The equation of this circle would be
x  y 1
2
2
(0,1)
(-1,0)
(1,0)
(0,-1)
Let's pick a point on the circle. We'll choose a point where the
x is 1/2. If the x is 1/2, what is the y value?
x  y 1
2
1
2

y
1
 
2
3
2
y 
4
(-1,0)
3
y
2
2
2
x = 1/2
(0,1)  1
3

2, 2 


(1,0)
1
3
 ,

2

2

(0,-1) 
Now, draw a right triangle from the origin to the point when x =
½. Use this triangle to find the values of sin, cos, and tan.
(0,1)
1
(-1,0)
1 3
 ,

2 2 


3
2

sin  
(1,0)
1
2
tan  
(0,-1)
cos
3
2  3
1
2
1
21
 1 2
3
2  3
1
2
For any point on the unit circle:
Sine is the y value
Cosine is the x value.
𝑦
Tangent is
𝑥
Find the sin, cos, and tan of the angle
 2 2


,
 2 2 


(-1,0)
(0,1)

1
3
 ,

2 2 


sin  
(1,0)
tan 
(0,-1)
1
3
 ,

2
2 


cos
2
2
2

2
2
 2  1
2

2
How many degrees are in each division?
0,1
 2 2


 2 , 2 


sin 225 
 2 2


 2 , 2 


90°
135°
45°
2

2
 1,0
180°
45°
0° 1,0
225°
 2
2


 2 , 2 


270°
0,1
315°
 2
2


,

 2
2 

How about radians?
0,1
 2 2


 2 , 2 


 1,0
7
sin

4
2
2
135°

180°
5
4225°
 2
2


,

 2
2 

 2 2


 2 , 2 


90°

3
4
2
 45°
4
3
2
270°
0° 1,0
7
4 315° 
2

0,1
 2

,
2

2 
How many degrees are in each division?
 1 3
 , 
 2 2 


cos 330  23
 3 1
 , 
 2 2


 1,0
0,1
120°
1 3
 , 
2 2 


90°
60°
150°
30°
180°
 3 1
 , 
 2 2


0° 1,0
30°
210°
 3 1


 2 , 2 


sin 240  
3
2
330°
240°
 1
3
  , 
 2 2 


270°
300°
0,1
 3 1
 , 
 2 2


1
3
 , 
2 2 


How about radians?
 1 3
 , 
 2 2 


120°
 3 1
 , 
 2 2


 1,0
0,1
1 3
 , 
2 2 


90°
60°
150°

30° 6
180°
30°
 3 1
 , 
 2 2


0° 1,0
210°
 3 1


 2 , 2 


330°
240°
 1
3
  , 
 2 2 


270°
300°
0,1
 3 1
 , 
 2 2


1
3
 , 
2 2 


1
3
 ,

2

2


Let’s think about the function f() = sin 
Domain:
Range:
All real numbers.
-1  sin   1
(0, 1)
(1, 0)
(-1, 0)
(0, -1)
Let’s think about the function f() = cos 
Domain:
Range:
All real numbers
-1  cos   1
(0, 1)
(-1, 0)
(1, 0)
(0, -1)
Let’s think about the function f() = tan 
Domain: x ≠ n( /2), where n is an odd integer
Range:
All real numbers
Trig functions:
Look at the unit circle and determine sin 420°.
Sine is periodic with a
period of 360° or 2.
1
3
 ,

2

2


So sin 420° = sin 60°.
Reciprocal functions have the same period.
PERIODIC PROPERTIES
sin( + 2) = sin 
cosec( + 2) = cosec 
cos( + 2) = cos 
sec( + 2) = sec 
tan( + ) = tan 
cot( + ) = cot 
9
tan
1
4
This would have the

same value as tan
Subtract the period until you reach
an angle measure you know.
4
Positive vs. Negative & Even vs. Odd
Even if: f(-x) = f(x)
Odd if: f(-x) = -f(x)
What is cos
1
2

3
?
 
What is cos   ?
 3
1
2
1
3
 ,

2

2


Remember negative angle means to go clockwise
cos    cos
What is sin

3
Cosine is an even function.
?
3
2
 
What is sin    ?
 3
3

2
1
3
 ,

2
2 

sin      sin 
What is tan

3
Sine is an odd function.
?
3
 
What is tan    ?
 3
 3
1
3
 ,

2
2 

EVEN-ODD PROPERTIES
sin(-  ) = - sin  (odd)
cosec(-  ) = - cosec  (odd)
cos(-  ) = cos  (even) sec(-  ) = sec  (even)
tan(-  ) = - tan  (odd)
cot(-  ) = - cot  (odd)
Problem Set 4.2