1. Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions

2. Characteristics of the cosecant Function
The graph is discontinuous at values of the form x = nπ and has vertical asymptotes at these values.
The domain is the set of all real numbers, except multiples of nπ where n is an integer.
The range is real numbers greater than/equal to 1 and less than/equal to -1
There are no x –intercepts.
The period is 2π.
The graph has no amplitude, since there is no maximum or minimum.

3. Characteristics of the cosecant Function
7. Graph is symmetric with respect to the origin, so the function is odd function. For all x in the domain, csc(-x) = - cscx

5. Characteristics of the secant Function
1) The graph is discontinuous at values of the form
x = π/2 + n π and has vertical asymptotes at these values.
2) The domain is the set of real numbers, except multiples of π/2 + n π where n is an integer.
3) The range is real numbers greater than/equal to 1 and less than/equal to -1.
4) There are no x –intercepts.
5) The period is 2π.

6. Characteristics of the secant Function
6. The graph has no amplitude, since there is no maximum or minimum.
7. Graph is symmetric with respect to the y-axis, so the function is an even function. For all x in the domain, sec(-x) = secx

8. Characteristics of the Tangent Function
The graph is discontinuous at values of the form x = π/ nπ and has vertical asymptotes at these values.
The domain is the set of real numbers except π/2 + nπ where n is an integer.
 The range consists of real numbers.
The tangent function is odd function (symmetric with respect to the origin).

9. Characteristics of the Tangent Function
5. The tangent function is periodic, with period π.
6. The x-intercepts are multiples of n π y, -intercept is 0.
7.The graph has no amplitude, since there is no maximum or minimum.

10. The Graph of y = f (x) = tan x

11. (0, 0)

12. y = tan x

13. Characteristics of the Cotangent Function
The graph is discontinuous at values of the form x = nπ and has vertical asymptotes at these values.
The domain is real numbers except nπ where n is an integer.
 The range consists of real numbers.
The cotangent function is odd function (symmetric with respect to the origin).

14. Characteristics of the Cotangent Function
5. The cotangent function is periodic, with period π.
6. The x-intercepts are of the form π/2 + nπ,
no y -intercept.
7. The graph has no amplitude, since there is no maximum or minimum.

18. y = tan (2x)

19. y = 3tan (2x)