Review : Solving Trig Equations
In this section we will take a look at solving trig
equations. This is something that you will be asked to do on a fairly regular
basis in my class.
Let�s just jump into the examples and see how to solve trig
equations.
Example Solve
 .
Solution
There�s really not a whole lot to do in solving this
kind of trig equation. All we need to do is divide both sides by 2 and the
go to the unit circle.
So, we are looking for all the values of t for
which cosine will have the value of
 .
So, let�s take a look at the following unit circle.
From quick inspection we can see that
 is
a solution. However, as I have shown on the unit circle there is another
angle which will also be a solution. We need to determine what this angle
is. When we look for these angles we typically want positive angles
that lie between 0 and
 .
This angle will not be the only possibility of course, but by convention we
typically look for angles that meet these conditions.
To find this angle for this problem all we need to do
is use a little geometry. The angle in the first quadrant makes an angle of
 with
the positive x-axis, then so must the angle in the fourth quadrant.
So we could use
 ,
but again, it�s more common to use positive angles so, we�ll use
 .
We aren�t done with this problem. As the discussion
about finding the second angle has shown there are many ways to write any
given angle on the unit circle. Sometimes it will be
 that
we want for the solution and sometimes we will want both (or neither) of the
listed angles. Therefore, since there isn�t anything in this problem
(contrast this with the next problem) to tell us which is the correct
solution we will need to list ALL possible solutions.
This is very easy to do. Recall from the previous
section and you�ll see there that I used
to represent all the possible angles that can end at
the same location on the unit circle, i.e. angles that end at
 .
Remember that all this says is that we start at
 then
rotate around in the counter-clockwise direction (n is positive) or
clockwise direction (n is negative) for n complete rotations.
The same thing can be done for the second solution.
So, all together the complete solution to this problem
is
As a final thought, notice that we can get
 by
using
 in
the second solution.
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