Trig Functions

Review : Trig Functions

The intent of this section is to remind you of some of the more important (from a Calculus standpoint�) topics from a trig class. One of the most important (but not the first) of these topics will be how to use the unit circle. We will actually leave the most important topic to the next section.

First let�s start with the six trig functions and how they relate to each other.

Recall as well that all the trig functions can be defined in terms of a right triangle.

From this right triangle we get the following definitions of the six trig functions.

Remembering both the relationship between all six of the trig functions and their right triangle definitions will be useful in this course on occasion.

Next, we need to touch on radians. In most trig classes instructors tend to concentrate on doing everything in terms of degrees (probably because it�s easier to visualize degrees). The same is true in many science classes. However, in a calculus course almost everything is done in radians. The following table gives some of the basic angles in both degrees and radians.

Know this table! We may not see these specific angles all that much when we get into the Calculus portion of these notes, but knowing these can help us to visualize each angle. Now, one more time just make sure this is clear.

Be forewarned, everything in most calculus classes will be done in radians!

Let�s next take a look at one of the most overlooked ideas from a trig class. The unit circle is one of the more useful tools to come out of a trig class. Unfortunately, most people don�t learn it as well as they should in their trig class.

Below is the unit circle with just the first quadrant filled in. The way the unit circle works is to draw a line from the center of the circle outwards corresponding to a given angle. Then look at the coordinates of the point where the line and the circle intersect. The first coordinate is the cosine of that angle and the second coordinate is the sine of that angle. We�ve put some of the basic angles along with the coordinates of their intersections on the unit circle. So, from the unit circle below we can see that and .

Remember how the signs of angles work. If you rotate in a counter clockwise direction the angle is positive and if you rotate in a clockwise direction the angle is negative.

Recall as well that one complete revolution is , so the positive x-axis can correspond to either an angle of 0 or (or , or , or , or , etc. depending on the direction of rotation). Likewise, the angle (to pick an angle completely at random) can also be any of the following angles:

(start at then rotate once around counter clockwise)

(start at then rotate around twice counter clockwise)

(start at then rotate once around clockwise)

(start at then rotate around twice clockwise)

etc.

In fact can be any of the following angles In this case n is the number of complete revolutions you make around the unit circle starting at . Positive values of n correspond to counter clockwise rotations and negative values of n correspond to clockwise rotations.

So, why did I only put in the first quadrant? The answer is simple. If you know the first quadrant then you can get all the other quadrants from the first with a small application of geometry.