1. Copyright © Cengage Learning. All rights reserved.5
Trigonometric Functions:
Unit Circle Approach
Copyright © Cengage Learning. All rights reserved.

2. Graphs of Tangent, Cotangent, Secant, and Cosecant
We begin by stating the periodic properties of these functions. We know that sine and cosine have period 2.
Since cosecant and secant are the reciprocals of sine and cosine, respectively, they also have period 2. Tangent and cotangent, however, have period .

3. Graphs of Tangent, Cotangent, Secant, and Cosecant
To see this, notice that as x gets close to  /2, cos x approaches 0 and sin x approaches 1 and so tan x = sin x/cos x is large.
A table of values of tan x for x close to  /2 ( ) is shown below. Thus the graph of y = tan x approaches the vertical lines x =  /2 and x = – /2. So these lines are vertical asymptotes.

4. Graphs of Tangent, Cotangent, Secant, and Cosecant
With the information we have so far, we sketch the graph of y = tan x for – /2 < x <  /2 in Figure 1.
One period of y = tan x
Figure 1

5. Graphs of Tangent, Cotangent, Secant, and Cosecant
The complete graph of tangent (see Figure 5(a)) is now obtained by using the fact that tangent is periodic with period .
y = tan x

6. Graphs of Tangent, Cotangent, Secant, and Cosecant
The function y = cot x is graphed on the interval (0,  ) by a similar analysis (see Figure 2)
Since y=1/tanx=cotx, whenever tanx has an output value of 0 (an x-intercept), the function cotx is undefined. Similarly, whenever tanx is undefined, cotx has an output value of 0. x
tanx
cotx
undefined
𝜋 2 𝜋
3𝜋 2
Undefined
One period of y = cot x

7. Graphs of Tangent, Cotangent, Secant, and Cosecant
Since cot x is undefined for x = n with n an integer, its complete graph (in Figure 5(b)) has vertical asymptotes at these values.
y = cot x

8. Graphs of Tangent, Cotangent, Secant, and Cosecant
To graph the cosecant and secant functions, we use the reciprocal identities
and
So to graph y = csc x, we take the reciprocals of the y-coordinates of the points of the graph of y = sin x. (See Figure 3.)
One period of y = csc x

9. Graphs of Tangent, Cotangent, Secant, and Cosecant
Similarly, to graph y = sec x, we take the reciprocals of the y-coordinates of the points of the graph of y = cos x. (See Figure 4.)
One period of y = sec x

10. Graphs of Tangent, Cotangent, Secant, and Cosecant
The complete graph in Figure 5(c) is now obtained from the fact that the function cosecant is periodic with period 2. Note that the graph has vertical asymptotes at the points where sin x = 0, that is, at x = n, for n an integer.
y = csc x

11. Graphs of Tangent, Cotangent, Secant, and Cosecant
The graph of y = sec x is sketched in a similar manner. Observe that the domain of sec x is the set of all real numbers other than x = ( /2) + n, for n an integer, so the graph has vertical asymptotes at those points. The complete graph is shown in Figure 5(d).
y = sec x

12. Graphs of Transformations of
Tangent and Cotangent

13. Graphs of Transformations of Tangent and Cotangent
To graph one period of y = a tan kx, an appropriate interval is
To graph one period of y = a cot kx, an appropriate interval is

14. Example 2 – Graphing Tangent Curves
Graph each function.
(a) y = tan 2x (b) y = tan 2

15. Example 3 – A Horizontally Shifted Cotangent Curve
Graph the function y = 2 cot

16. Graphs of Transformations of Cosecant and Secant

17. Graphs of Transformations of Cosecant and Secant
An appropriate interval on which to graph one complete period is [0, 2 /k].

18. Example 4 – Graphing Cosecant Curves
Graph each function.

19. Example 5 – Graphing a Secant Curve
Graph y = 3 sec

20. Example 6
Write a function for the following graph.