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.
|
Degree
|
0
|
30
|
45
|
60
|
90
|
180
|
270
|
360
|
|
Radians
|
0
|
|
|
|
|
|
|
|
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. You’ll see how this is done in
the following set of examples.
|
Example 1 Evaluate
each of the following.
(a) and [Solution]
(b) and [Solution]
(c) and [Solution]
(d) [Solution]
Solution
(a) The first evaluation in this part uses the angle . That’s not on our unit circle above,
however notice that . So is found by rotating up from the negative x-axis. This means that
the line for will be a mirror image of the line for only in the second quadrant. The coordinates for will be the coordinates for except the x coordinate will be negative.
Likewise for we can notice that ,
so this angle can be found by rotating down from the negative x-axis. This means that the
line for will be a mirror image of the line for only in the third quadrant and the
coordinates will be the same as the coordinates for except both will be negative.
Both of these angles along with their coordinates are
shown on the following unit circle.
From this unit circle we can see that and .
This leads to a nice fact about the sine function. The sine function is called an odd function and so for ANY angle we
have
[Return to Problems]
(b) For this example notice that so this means we would rotate down from the negative x-axis to get to this angle. Also so this means we would rotate up from the negative x-axis to get to this angle.
So, as with the last part, both of these angles will be mirror images
of in the third and second quadrants
respectively and we can use this to determine the coordinates for both of
these new angles.
Both of these angles are shown on the following unit
circle along with appropriate coordinates for the intersection points.
From this unit circle we can see that and . In this case the cosine function is called
an even function and so for ANY
angle we have
.
[Return to Problems]
(c) Here we should note that so and are in fact the same angle! Also note that this angle will be the
mirror image of in the fourth quadrant. The unit circle for this angle is
Now, if we remember that we can use the unit circle to find the
values of the tangent function. So,
.
On a side note, notice that and we can see that the tangent function is
also called an odd function and so
for ANY angle we will have
.
[Return to Problems]
(d) Here we need to notice that . In other words, we’ve started at and rotated around twice to end back up at
the same point on the unit circle.
This means that
Now, let’s also not get excited about the secant
here. Just recall that
and so all we need to do here is evaluate a cosine! Therefore,
[Return to Problems]
|
So, in the last example we saw how the unit circle can be
used to determine the value of the trig functions at any of the “common”
angles. It’s important to notice that
all of these examples used the fact that if you know the first quadrant of the
unit circle and can relate all the other angles to “mirror images” of one of
the first quadrant angles you don’t really need to know whole unit circle. If you’d like to see a complete unit circle
I’ve got one on my Trig Cheat Sheet
that is available at http://tutorial.math.lamar.edu.
Another important idea from the last example is that when it
comes to evaluating trig functions all that you really need to know is how to
evaluate sine and cosine. The other four
trig functions are defined in terms of these two so if you know how to evaluate
sine and cosine you can also evaluate the remaining four trig functions.
We’ve not covered many of the topics from a trig class in
this section, but we did cover some of the more important ones from a calculus
standpoint. There are many important
trig formulas that you will use occasionally in a calculus class. Most notably are the half-angle and
double-angle formulas. If you need
reminded of what these are, you might want to download my Trig Cheat Sheet
as most of the important facts and formulas from a trig class are listed there.