Trig Function Evaluation
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One of the problems with most trig classes is that they tend
to concentrate on right triangle trig and do everything in terms of
degrees. Then you get to a calculus
course where almost everything is done in radians and the unit circle is a very
useful tool.
So first off let’s look at the following table to relate
degrees and radians.
Degree

0

30

45

60

90

180

270

360

Radians

0








Know this table!
There are, of course, many other angles in radians that we’ll see during
this class, but most will relate back to these few angles. So, if you can deal with these angles you
will be able to deal with most of the others.
Be forewarned,
everything in most calculus classes will be done in radians!
Now, let’s look at the unit circle. 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. There are a couple of basic angles that are commonly used. These are and are shown below along with the coordinates
of the intersections. 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 xaxis 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.
You’ll see this in the following examples.
Find the exact value of each of the following. In other words, don’t use a calculator.
1. and
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2. and
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3. and
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4.
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5.
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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,
We should also note that cosine
and secant are periodic functions with a period of . So,
6.
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Trig Evaluation Final
Thoughts
As we saw in the previous examples if you know the first
quadrant of the unit circle you can find the value of ANY trig function (not
just sine and cosine) for ANY angle that can be related back to one of those
shown in the first quadrant. This is a
nice idea to remember as it means that you only need to memorize the first
quadrant and how to get the angles in the remaining three quadrants!
In these problems I used only “basic” angles, but many of
the ideas here can also be applied to angles other than these “basic” angles as
we’ll see in Solving Trig Equations.