Other Circular Functions

Objectives:

Identify the other basic circular functions (besides sine and cosine)
Compute circular functions of any real number with your calculator
For the circular functions:
- Be able to graph each
- Recognize each as a periodic function
  f(t + p) = f(t) where p the smallest possible positive value which makes the equation true for all t in the domain of f
- Identify the graph of each when given the equation f(x) = A ???[C(x - D)] + B (where ??? is sin,cos,tan,csc,sec or cot)
- Identify the properties of their graphs
  - Period (the value of p from above or |2pi/C| or |pi/C| for tan and cot )
  - Amplitude (half the total height if finite, no amplitude if the height is infinite)
  - Domain (watch for vertical asymptotes)
  - Range

Definition of Circular Functions:

For any real number t (considered as a length), determine a point (x,y) on the unit circle. Note that the length t of an arc on the unit circle is exactly the same as the radian measure of the angle since arc length is equal to the radius times the angle in radians and in this case the radius is one. Then we have these functions:

Circular Functions
sin t = y
cos t = x
tan t = y/x
csc t = 1/y
sec t = 1/x
cot t = x/y

Basic Circular Function Facts:

Complete the following table:

Circular Function Domain Range Period Amplitude

Graphs of the Six Basic Circular Functions:

Graphs of the Six Basic Circular Functions

Defining New Functions With the TI-89:

You can use the TI-89 to define the cosecant, cotangent, and secant functions which are not already in your calculator. To define the secant function, type the following into the TI-89 display:

Define sec(x)=1/cos(x) [ENTER]
To use the function, type the following into the TI-89 display:

2nd VAR-LINK
arrow to the line which says sec
type in 60) [ENTER] in degree mode

You should get 2 for the answer in this case. Proceed similarly with definitions for cot(x) and csc(x).

Food For Thought:

Which two of the six basic circular functions have an amplitude?
Which two of the six basic circular functions have a period of pi?
Which two of the six basic circular functions have no roots?