The Tangent Function
Slope on the Unit Circle
1
(cosӨ,sinӨ)
Ө
-1
cosӨ
sinӨ
What is the slope of
the terminal side of
an angle on the unit
circle?
Opposite
1
Adjacent
Using our
knowledge of
the Unit Circle…
sin  
Or using
trigonometry…
opposite
 tan  
Slope = cos   -1
adjacent
A Definition of Tangent
The tangent function is defined as:
tan   
sin  
cos  
There are values for which the tangent function are undefined:
Any Θ that makes cos(Θ)=0.
  2,

3
2
,
5
2
,
7
2
,
9
2
,
11
2
,...
Example
Find the exact value of the following:
Thought process
tan  53 

Reference Angle:
3
Tangent of Reference Angle: tan  3   3
Quadrant of Angle: 5  60  300  Fourth Quadrant
Sign of Tangent in Fourth Quadrant:
 ,  
 negative
positive  Negative
 , 
Therefore:
tan  53    3
 ,  
 , 
The only thing required for a correct
answer (unless the question says explain)
The Tangent Function Graph
In order to investigate the
tangent function, first
examine all the values of
sine and cosine. Remember,
tangent is sine divided by
cosine.
Now find and graph all of
the values of sine÷cosine.
X
SIN(X)
COS(X)
-2π
0
1
-7π/4
0.707
0.707
-3π/2
1
0
-5π/4
0.707
-0.707
-π
0
-1
-3π/4
-0.707
-0.707
-π/2
-1
0
-π/4
-0.707
0.707
0
0
1
π/4
0.707
0.707
π/2
1
0
3π/4
0.707
-0.707
π
0
-1
5π/4
-0.707
-0.707
3π/2
-1
0
7π/4
-0.707
0.707
2π
0
1
The Tangent Function Graph


Plot the points. The errors are asymptotes.
X
SIN(X)
COS(X)
TAN(X)
-2π
0
1
0/1 = 0
-7π/4
0.707
0.707
.707/.707 = 1
-3π/2
1
0
1/0 = DNE
-5π/4
0.707
-0.707
.707/-.707 = -1
-π
0
-1
0/-1 = 0
-3π/4
-0.707
-0.707
-.707/-.707 = 1
-π/2
-1
0
-1/0 = DNE
-π/4
-0.707
0.707
-.707/.707 = -1
0
0
1
0/1 = 0
π/4
0.707
0.707
.707/.707 = 1
π/2
1
0
1/0 = DNE
3π/4
0.707
-0.707
.707/-.707 = -1
π
0
-1
0/-1 = 0
5π/4
-0.707
-0.707
-.707/-.707 = 1
3π/2
-1
0
-1/0 = DNE
7π/4
-0.707
0.707
-.707/.707 = -1
2π
0
1
0/1 = 0
Find the values of sine divided by cosine.
y  tan  x 
The Tangent Function Graph
y  tan  x 
Domain:
All Reals except
...  32 ,  2 , 2 , 32 ,...


Range:
All Reals
Asymptotes
x  ... 
3
2
, 2 , 2 ,


3
2
,...
Graph of Tangent
(For 0 ≤ x ≤ 2π)
Asymptotes: x  2 , 32
Domain: 0  x  2

2

3
2
except x  2 , 32
2 Range: All Reals
x-intercepts: 0,  , 2
y -intercept:  0, 0 
Length of Cycle: