1. Warm-up: HW: Graph Tangent and Cotangent Functions
Complete the table from θ = −/2 to θ = /2 (one  interval), with the given 5 input angle values. x
sinx
cosx
y = tanx
−/2
−/4
/4
/2 −1
und −1 1 1 1
und
HW: Graph Tangent and Cotangent Functions

2. Objective:
Graph tangent and cotangent functions.

3. Tangent Function Recall that when cos θ = 0, tan θ is undefined.
This occurs at  intervals, offset by /2: { … − /2, /2, 3/2, 5/2, … }
From the warm-up we can create a t-table for y = tanx x
y = tanx
−/2
und
−/4 −1
/4 1
/2 x
sinx
cosx
y = tanx
−/2
−/4
/4
/2 −1
und −1 1 1 1
und

4. We are interested in the graph of y = tan x
vertical asymptote
x y = tan x x

5. Graph of the Tangent Functiony x
Properties of y = tan x
1. Domain : all real x
2. Range: (–, +)
3. Period: /ω
4. Vertical asymptotes:
period:

6. Transformations apply as usual. Let’s try one.
up 2
reflect about x-axis
right /4

7. Finding Domain and Range:
Domain for y = tanx:
all real x
For
2. Range: (–, +)

8. Finding the Period:
Since the period of tangent is , the period of tan x is:
The period would be /2
y = tan x
y = tan 2x y x y x
period:
period:

9. Graphing Variations of y = tan x
¼ and ¾ distance from asymptote to asymptote
Properties of y = tan x
1. Domain : all real x
2. Range: (–, +)
3. Period: /ω
4. Vertical asymptotes:

10. Example: Graphing a Tangent Function
¼ and ¾ distance from asymptote to asymptote
Graph y = 3 tan 2x
Step 1: Find two asymptotes.
An interval containing one period is or /ω = /2

11. x = 0 is midway between and The graph passes through (0, 0).
Graph y = 3 tan 2x
Step 2: Identify an x-intercept, midway between the consecutive asymptotes.
x = 0 is midway between and
The graph passes through (0, 0).
¼ and ¾ distance from asymptote to asymptote

12. The graph passes through and
¼ and ¾ distance from asymptote to asymptote
Graph y = 3 tan 2x
Step 3: Find points on the graph 1/4 and 3/4 of the way between the consecutive asymptotes. These points have
y-coordinates of –A and A.
The graph passes through
and

13. y = 3 tan 2x Finding Domain and Range: Properties of y = tan x
1. Domain : all real x
2. Range: (–, +)
3. Period: /ω
4. Vertical asymptotes:
Properties for y = 3tan2x
Range: (–, +)
Vertical asymptotes:
Domain:

14. Cotangent Function Recall that . when sin θ = 0, cot θ is undefined.
This occurs at  intervals, starting at 0: { … −, 0, , 2, … }
Let’s create a t-table from θ = 0 to θ =  (one  interval),
with 5 input angle values. θ
cot θ
Und
/4 1
/2
3/4 −1  θ
sin θ
cos θ
cot θ
/4
/2
3/4  1
Und 1 1 −1 –1
Und

15. Graph of Cotangent Function: Periodic
Vertical asymptotes
where sin θ = 0
cot θ θ
cot θ
und
/4 1
/2
3/4 −1 
−3/2
- 
− /2
/2 
3/2

16. Graph of the Cotangent Function
To graph y = cot x, use the identity
At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes. y x
Properties of y = cot x
vertical asymptotes
1. Domain : all real x
2. Range: (–, +)
3. Period: /ω
4. Vertical asymptotes:

17. Graphing Variations of y = cot x
Properties of y = cot x
1. Domain : all real x
¼ and ¾ distance from asymptote to asymptote
2. Range: (–, +)
3. Period: /ω
4. Vertical asymptotes:

18. Example: Graphing a Cotangent Functiony x

19. x =  is midway between x = 0 and x = 2.
Graph
Step 2 Identify an x-intercept midway between the consecutive asymptotes.
x =  is midway between x = 0 and x = 2.
The graph passes through (, 0). y x

20. The graph passes through and
Step 3 Find points on the graph 1/4 and 3/4 of the way between consecutive asymptotes. These points have y-coordinates of A and –A.
The graph passes through
and y x

21. Finding Domain and Range:
Properties for
Properties of y = cot x
Domain:
1. Domain : all real x
Range: (–, +)
2. Range: (–, +) y x
3. Period: /ω
4. Vertical asymptotes:
Vertical asymptotes:

22. ¼ and ¾ distance from asymptote to asymptote

23. Sneedlegrit:
HW: Graph Tangent and Cotangent Functions