There are six basic trigonometric functions that we will use over and over throughout the course. There are two ways that we can define these functions, each having it's own advantages and disadvantages. The first way that we will look at the trig functions is as points on the unit circle.
Unit Circle Trigonometry
First, we need to understand the unit circle. As the name implies, it is a circle where the radius is 1 (one unit). The unit circle equation is x² + y² = 1 and the graph looks like this:
Now, if we start at the point (1, 0) and walk a distance t around the circle, we will arrive at a point (x, y) represented by the blue point on the circle in the figure below. The distance traveled, t, is shown in red. At the right of the circle is a red line segment that ends in a blue point that is exactly the same length as the red arc ending in the blue point.
We are interested in the (x, y) coordinates of the point corresponding to going around the circle a distance of t.
There are six trigonometric functions: sine (abbreviated sin), cosine (abbreviated cos), tangent (abbreviated tan), cosecant (abbreviated csc), secant (abbreviated sec), and cotangent (abbreviated cot).
Here's how the trigonometric functions are defined. Let t be the distance traveled around the unit circle ending at the point (x, y):
sin(t) = y,
cos(t) = x,
tan(t) = y/x so long as x is not 0,
csc(t) = 1/y, so long as y is not 0,
sec(t) = 1/x so long as x is not 0,
cot(t) = x/y so long as y is not 0.
Note that csc is the reciprocal of sin, sec is the reciprocal of cos, and cot is the reciprocal of tan.
Here is an interactive website that helps you explore the relationship between the (x, y) point on the circle and the value of the sine, cosine, tangent, etc.: Six Trig Functions. To use the applet, move the mouse over the red dot (point) on the circle. Then click and drag the point around the circle to see the measure of the angle change as well as the sin & cos. You can have the applet display the other trig functions (cot, csc, etc.) as well.
A little mneumonic (memory aid) that I use to keep sine and cosine straight is that x comes before y in the alphabet and cosine comes before sine in the dictionary. Therefore the cos(t) = x and sin(t) = y. If it helps you remember, please feel free to use it.
There are some important angles to know by heart (memorize). They are summarized in the figure below. Here's how to read the figure. Look at the point corresponding to t = π/3. The x-coordinate is 1/2 = 0.5 and the y-coordinate is sqrt(3)/2. Therefore cos(π/3) = 0.5 and the sin(π/3) = sqrt(3)/2.
PLEASE memorize each of these major (important) angles and their coordinates (cosine & sine). It will serve you well and save you lots of time, especially when we get to inverse functions later on in chapter 1.
Supplemental Video Clips
I have found some video clips online (from University of Idaho) that help further explain and give more examples for the trigonometry concepts we've learned in this lesson. While they are optional, many of them are recommended to help you solidify your understanding of the concepts. Note that you will need to have a pretty good internet connection for these to work properly.
To view these clips, you will need to have the Real Player program installed on your computer. If you don't already have Real Player, you can download it to your PC (for free) by visiting this web page: Real Player download page. Just click on the button that says, "Start RealPlayer Download" and follow the instructions. (It will probably offer you the chance to purchase additional programs like Rhapsody, etc., but I don't recommend it because you will have to pay for those out of your own pocket).
Once Real Player is installed on your computer you may view the following video clips. After the name of each video clip is the length of the video in minutes and seconds. Example (7:12) is a video clip that is 7 minutes and 12 seconds long.
Video Clips related to Unit Circle Trigonometry:
The Unit Circle (7:12) *Strongly recommended that you watch this one*
Given an angle, find the Corresponding Points on the Unit Circle (12:06)
Evaluating the Trig Ratios Using the Unit Circle (6:53) *Strongly recommended*
Evaluating the Trig Ratios at 0,90,180,270,360 (6:25) *Strongly recommended*
Evaluating the Trig Ratios at Increments of 45 Degrees (9:35) *Recommended*
Evaluating the Trig Ratios at Increments of 30 Degrees (10:45) *Recommended*
Evaluating the Trig Ratios at Increments of 60 Degrees (4:16) *Recommended*