Sum and Difference Identities

1
Sum and Difference Identities

• Section 5.2

2
Objectives

• Apply a sum or difference identity to evaluate
  the sine or cosine of an angle.

3
Sum and Difference Identities
The identity above is a short hand method for
writing two identities as one. When these
identities are broken up, they look like
The identity above is a short hand method for
writing two identities as one. When these
identities are broken up, they look like
4
Use a sum or difference identity to find the
exact value of
In order to answer this question, we need to find
two of the angles that we know to either add
together or subtract from each other that will
get us the angle p/12. Lets start by looking at
the angles that we know
continued on next slide
5
Use a sum or difference identity to find the
exact value of
We have several choices of angles that we can
subtract from each other to get p/12. We will
pick the smallest two such angles
Now we will use the difference formula for the
sine function to calculate the exact value.
continued on next slide
6
Use a sum or difference identity to find the
exact value of
For the formula a will be
and b will be
This will give us
continued on next slide
7
Use a sum or difference identity to find the
exact value of
For the formula a will be
and b will be
This will give us
8
Simplify using a sum or difference identity
In order to answer this question, we need to use
the sine formula for the sum of two angles.
For the formula a will be
and b will be
continued on next slide
9
Simplify using a sum or difference identity
10
Simplify using a sum or difference identity
In order to answer this question, we need to use
the cosine formula for the difference of two
angles.
For the formula a will be
and b will be
continued on next slide
11
Simplify using a sum or difference identity
12
Find the exact value of the following
trigonometric functions below given
and
For this problem, we have two angles. We do not
actually know the value of either angle, but we
can draw a right triangle for each angle that
will allow us to answer the questions.
continued on next slide
13
Find the exact value of the following
trigonometric functions below given
and
Triangle for a
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14
Find the exact value of the following
trigonometric functions below given
and
Triangle for ß
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15
Find the exact value of the following
trigonometric functions below given
and
Now that we have our triangles, we can use the
cosine identity for the sum of two angles to
complete the problem.
continued on next slide
16
Find the exact value of the following
trigonometric functions below given
and
Note Since a is in quadrant Iv, the sine value
will be negative
Now that we have our triangles, we can use the
cosine identity for the sum of two angles to
complete the problem.
Note Since ß is in quadrant II, the cosine value
will be negative
continued on next slide
17
Find the exact value of the following
trigonometric functions below given
and
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18
Find the exact value of the following
trigonometric functions below given
and
While we are here, what are the possible
quadrants in which the angle aß can fall?
In order to answer this question, we need to know
if cos(aß) is positive or negative. We can type
the value into the calculator to determine this.
When we do this, we find that cos(aß) is
positive. The cosine if positive in quadrants I
and IV. Thus aß must be in either quadrant I or
IV. We cannot narrow our answer down any further
without knowing the sign of sin(aß).
continued on next slide
19
Find the exact value of the following
trigonometric functions below given
and
Note Since a is in quadrant Iv, the sine value
will be negative
Now that we have our triangles, we can use the
cosine identity for the sum of two angles to
complete the problem.
Note Since ß is in quadrant II, the cosine value
will be negative
continued on next slide
20
Find the exact value of the following
trigonometric functions below given
and