Trigonometric Equations

1
Trigonometric Equations

• Section 5.5

2
Objectives

• Solve trigonometric equations.

3
Solve the equation on the interval
This question is asking What angle(s) on the
interval 0, 2p) have a sine value of ?
This is a sine value that we should recognize as
one of our standard angle on the unit circle.
Thus we need do no work, but instead just answer
the question from memory.
4
Solve the equation on the interval
This question is asking What angle(s) on the
interval 0, 2p) have a cosine value of
?
This is a sine value that we should recognize as
one of our standard angle on the unit circle.
Thus we need do no work, but instead just answer
the question from memory.
5
Solve the equation on the interval
Although we recognize the ½ as a value we know,
since the sine function is squared, we first must
take the square root of both sides of the
equation.
continued on next slide
6
Solve the equation on the interval
What we have now is really two equations to solve.
These are sine values that we should recognize as
some of our standard angles on the unit circle.
Thus we need do no work, but instead just answer
the question from memory.
For the left equation
For the right equation
7
Solve the equation on the interval
For this problem, we must do some algebraic work
to get to an equation like the previous ones.
Here we need to factor. This equation can be
factored by grouping.
The grouping can be seen here with the red and
blue boxes. The part surrounded by the red box
has a cos(t) that can be factored out of both
pieces.
Now you should notice that the part in the square
brackets and the part in the blue box are the
same. This means that we can factor this part
out.
continued on next slide
8
Solve the equation on the interval
Now this is really two equations that need to be
solved
If we solve each of these equations so that the
trigonometric function is on the left and all the
numbers are on the right, it will looks just like
our previous problems.
or
continued on next slide
9
Solve the equation on the interval
Each of these equations, we should know the
answers to.
or
or
10
Solve the equation on the interval
In order to do this problem, we need to angles to
be the same. The angle in the sine function is
2x. The angle in the cosine function is x.
Since we have an identity for the double angle of
a sine function, we can replace sin(2x) with the
identity. This will give us the same angles in
all of the trigonometric functions.
Now we can factor a cos(x) out of each piece to
get
continued on next slide
11
Solve the equation on the interval
In order to do this problem, we need to angles to
be the same. The angle in the sine function is
2x. The angle in the cosine function is x.
Since we have an identity for the double angle of
a sine function, we can replace sin(2x) with the
identity. This will give us the same angles in
all of the trigonometric functions.
Now we can factor a cos(x) out of each piece to
get
continued on next slide
12
Solve the equation on the interval
This is really two equations to solve
The equation on the left, we should know the
solutions to. The equation on the right, we can
solve with some manipulation.
or
13
Solve the equation on the interval
Our first step in the problem will be to take the
square root of both sides of the equation.
This will give us two equations to solve.
or
Neither of these is a value that we know from our
standard angles.
continued on next slide
14
Solve the equation on the interval
In order to find these angles, we will need to
use our inverse trigonometric functions.
or
These are not all of the answers to the question.
Lets start by looking at the left side.
This angle is between 0 and p/2 in quadrant I.
We know this because the range of the inverse
sine function is
This means that this value for t is one of our
answer in the interval that we need. We are not,
however, getting all of the angles where the sine
value is ¼ .
continued on next slide
15
Solve the equation on the interval
How do we find the other angles? We use
reference angles and what we know about
quadrants. All of the angles that have a sine
value of ¼ will have the same reference angle.
What is this reference angle? In this case, the
reference angle is
since this angle is in quadrant I and all angles
in quadrant I and in the interval 0, 2p) is its
own reference angle.
The next question is What other quadrant will
have a positive sine value? The answer to this
question is quadrant II. In quadrant II
reference angles are found as follows
continued on next slide
16
Solve the equation on the interval
We can use this formula to find the angle in
quadrant II since we know the reference angle.
We will just plug in the reference angle and
solve for the angle in quadrant II.
continued on next slide
17
Solve the equation on the interval
This gives us the following two values for the
solution to
or
Now we will work on the other equation that was
on the right in a previous slide.
continued on next slide
18
Solve the equation on the interval
In order to find these angles, we will need to
use our inverse trigonometric functions.
or
These are not all of the answers to the question.
We will continue by looking at the right side
This angle is between -p/2 and 0 in quadrant IV.
We know this because the range of the inverse
sine function is
This means that this value for t is not one of
our answer in the interval that we need. We need
to do a bit more work to get to the answers.
continued on next slide
19
Solve the equation on the interval
One thing that we can do to find an angle in the
interval 0, 2p), it to find an angle coterminal
to that is in the
interval 0, 2p). We do this by adding 2p to
the angle we have.
or
This is one angle that fits our criteria. How do
we find all other angles on the interval 0, 2p)
that also have a sine value of -¼ ?
continued on next slide
20
Solve the equation on the interval
How do we find the other angles? We use
reference angles and what we know about
quadrants. All of the angles that have a sine
value of -¼ will have the same reference angle.
What is this reference angle? In this case, we
will find the reference angle for the one angle
that we have in quadrant IV.
continued on next slide
21
Solve the equation on the interval
The next question is What other quadrant will
have a negative sine value? The answer to this
question is quadrant III. In quadrant III
reference angles are found as follows
We can use this formula to find the angle in
quadrant III since we know the reference angle.
We will just plug in the reference angle and
solve for the angle in quadrant III.
continued on next slide
22
Solve the equation on the interval
This gives us the following two values for the
solution to
or
Thus we have all solutions to the original
equation.
or
or
23
Find all solutions to the equation
Once factored, this is really two equations to
solve
The equation on the left, we should know the
solutions to. The equation on the right, we can
solve with some manipulation.
or
24
Solve the equation on the interval
This equation is a quadratic equation in tan(x).
Since it is not easily seen to be factorable, we
can use the quadratic formula to make a start on
finding the solutions to this equation.
continued on next slide
25
Solve the equation on the interval
This is really two equations to solve.
We are going to go through the same process that
we did with the previous problem to find the
answers.
continued on next slide
26
Solve the equation on the interval
In order to find these angles, we will need to
use our inverse trigonometric functions.
or
These are not all of the answers to the question.
Lets start by looking at the left side.
This angle is between 0 and p/2 in quadrant I.
We know this because the range of the inverse
tangent function is
This means that this value for t is one of our
answer in the interval that we need. We are not,
however, getting all of the angles where the
tangent value is 1.6 .
continued on next slide
27
Solve the equation on the interval
How do we find the other angles? We use
reference angles and what we know about
quadrants. All of the angles that have a tangent
value of 1.6 will have the same reference angle.
What is this reference angle? In this case, the
reference angle is
since this angle is in quadrant I and all angles
in quadrant I and in the interval 0, 2p) is its
own reference angle.
The next question is What other quadrant will
have a positive tangent value? The answer to
this question is quadrant III. In quadrant III
reference angles are found as follows
continued on next slide
28
Solve the equation on the interval
We can use this formula to find the angle in
quadrant II since we know the reference angle.
We will just plug in the reference angle and
solve for the angle in quadrant II.
continued on next slide
29
Solve the equation on the interval
This gives us the following two values for the
solution to
or
Now we will work on the other equation that was
on the right in a previous slide.
continued on next slide
30
Solve the equation on the interval
In order to find these angles, we will need to
use our inverse trigonometric functions.
or
These are not all of the answers to the question.
We will continue by looking at the right side
This angle is between -p/2 and 0 in quadrant IV.
We know this because the range of the inverse
tangent function is
This means that this value for t is not one of
our answer in the interval that we need. We need
to do a bit more work to get to the answers.
continued on next slide
31
Solve the equation on the interval
One thing that we can do to find an angle in the
interval 0, 2p), it to find an angle coterminal
to that is in the
interval 0, 2p). We do this by adding 2p to
the angle we have.
This is one angle that fits our criteria. How do
we find all other angles on the interval 0, 2p)
that also have a tangent value of -2.5 ?
continued on next slide
32
Solve the equation on the interval
How do we find the other angles? We use
reference angles and what we know about
quadrants. All of the angles that have a tangent
value of -2.5 will have the same reference angle.
What is this reference angle? In this case, we
will find the reference angle for the one angle
that we have in quadrant IV.
continued on next slide
33
Solve the equation on the interval
The next question is What other quadrant will
have a negative tangent value? The answer to
this question is quadrant II. In quadrant II
reference angles are found as follows
We can use this formula to find the angle in
quadrant II since we know the reference angle.
We will just plug in the reference angle and
solve for the angle in quadrant II.
continued on next slide
34
Solve the equation on the interval
This gives us the following two values for the
solution to
or
Thus we have all solutions to the original
equation.
or
or
continued on next slide
35
Solve the equation on the interval
36
Solve the equation on the interval