MTH 120 — Spring — 2008
Essex County College — Division of Mathematics
Handout — Version 41 — January 6, 2008
1
Trigonometric Functions of Real Numbers
The definitions of the trigonometric functions will not change, but now will will use thetas that
aren’t so nice.2 Before we just used values of theta that were easy to evaluate. Here’s some
questions that you should be able to answer now.
1. Are there any angles where sine and cosine are undefined? If so, what are they?
2. Are there any angles where the other four trigonometric functions are undefined? If so,
what are they?
3. Considering the angles from 0 to 2π, what angles should you be able to evaluate (or state
that they’re undefined) the trigonometric functions.
1
This document was prepared by Ron Bannon using LATEX 2ε .
Any angle that is co-terminal with the coordinate axis, or those angles whose reference is 30◦ , 45◦ , or 60◦ are
considered nice.
2
1
4. What is the domain and range of each of the six trigonometric functions?
1.1
Calculator Usage
You should be able to use your calculator to evaluate the trigonometric functions. Clearly you
should see the sine, cosine and tangent functions on your calculator. However you probably
don’t see cotangent, secant and cosecant on your calculators, but you should know that these
three functions are related to sine, cosine and tangent.
cot θ =
1
tan θ
sec θ =
1
cos θ
sec θ =
1
sin θ
You should also be aware that the angle theta can be given in various ways, and your calculator
has at least two modes: radian and degree. Make sure that you know how to move back and
forth between radian and degree mode.
2
Examples
For the given theta, evaluate each of the six trigonometric function to four decimal places.
1. θ = 25◦
Work:
2
2. θ = 25
Work:
3. θ = π ◦
Work:
4. θ = 17.289◦
Work:
5. θ = 0.26859
Work:
3
3
Graphing
I hope you recall the topic of translation from MTH-119, where you had many simple graphs and
you were asked to draw a related graph using simple translation rules. Methods may vary, but
you nonetheless were given a parent, f (x), and then asked to draw the child, A·f (Bx ± C)±D.
The same will be true for the trigonometric functions, and what follows will be parent graphs
of all six trigonometric functions. For now we will discuss key points on each of these curves.
Although there’s no scale, we will label each graph properly in class. Once these six graphs are
understood we will be able to easily translate these graphs exactly as was done in MTH-119.
0
Figure 1: Partial graph of f (θ) = sin θ.
0
Figure 2: Partial graph of f (θ) = cos θ.
4
0
Figure 3: Partial graph of f (θ) = tan θ.
0
Figure 4: Partial graph of f (θ) = cot θ.
5
0
Figure 5: Partial graph of f (θ) = sec θ.
0
Figure 6: Partial graph of f (θ) = csc θ.
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