1. The Tangent Function

2. Slope on the Unit Circle
What is the slope of the terminal side of an angle on the unit circle? 1
(cosӨ,sinӨ)
sinӨ
Opposite Ө -1
cosӨ 1
Adjacent
Using our knowledge of the Unit Circle…
Or using trigonometry…
Slope = -1

3. A Definition of Tangent
The tangent function is defined as:
There are values for which the tangent function are undefined:
Any Θ that makes cos(Θ)=0.

4. Example Find the exact value of the following: Thought process
The only thing required for a correct answer (unless the question says explain)

5. The Tangent Function GraphX
SIN(X)
COS(X)
-2π 1
-7π/4
0.707
-3π/2
-5π/4
-0.707 -π -1
-3π/4
-π/2
-π/4
π/4
π/2
3π/4 π
5π/4
3π/2
7π/4 2π
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.

6. The Tangent Function GraphX
SIN(X)
COS(X)
-2π 1
-7π/4
0.707
-3π/2
-5π/4
-0.707 -π -1
-3π/4
-π/2
-π/4
π/4
π/2
3π/4 π
5π/4
3π/2
7π/4 2π
TAN(X)
0/1 = 0
.707/.707 = 1
1/0 = DNE
.707/-.707 = -1
0/-1 = 0
-.707/-.707 = 1
-1/0 = DNE
-.707/.707 = -1
Find the values of sine divided by cosine.
Plot the points.
The errors are asymptotes.

7. The Tangent Function Graph
Domain: Range: Asymptotes
All Reals except
All Reals

8. Graph of Tangent
(For 0 ≤ x ≤ 2π)