Sum and Difference Identities Section 5.2
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Sum and Difference Identities
Section 5.2
Objectives• Apply a sum or difference identity to
evaluate the sine or cosine of an angle.
Sum and Difference Identities)cos()sin()cos()sin()sin( abbaba
)sin()sin()cos()cos()cos( bababa
The identity above is a short hand method for writing two identities as one. When these identities are broken up, they look like
)cos()sin()cos()sin()sin( abbaba
)sin()sin()cos()cos()cos( bababa
)cos()sin()cos()sin()sin( abbaba
)sin()sin()cos()cos()cos( bababa
The identity above is a short hand method for writing two identities as one. When these identities are broken up, they look like
Use a sum or difference identity to find the exact
value of
12
12
12
sin
12
9
4
3
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 π/12. Let’s start by looking at the angles that we know:
12
10
6
5
122
6
123
4
12
8
3
2
126
2
124
3
continued on next slide
Use a sum or difference identity to find the exact
value of
12
sin
We have several choices of angles that we can subtract from each other to get π/12. We will pick the smallest two such angles:
122
6
123
4
continued on next slide
Now we will use the difference formula for the sine function to calculate the exact value.
)cos()sin()cos()sin()sin( abbaba
Use a sum or difference identity to find the exact
value of
12
sin
For the formula a will be 12
26
123
4
continued on next slide
22
21
23
22
12sin
4cos
6sin
6cos
4sin
64sin
122
123
sin12
sin
and b will be
This will give us
Use a sum or difference identity to find the exact
value of
12
sin
For the formula a will be 12
26
123
4
4
13212
sin
4232
12sin
42
432
12sin
and b will be
This will give us
Simplify
using a sum or difference identity
4sin
x
)cos()sin()cos()sin()sin( abbaba
In order to answer this question, we need to use the sine formula for the sum of two angles.
continued on next slide
For the formula a will be 4
x and b will be
)cos(4
sin4
cos)sin(4
sin xxx
Simplify
using a sum or difference identity
4sin
x
)cos()sin(22
4sin
)cos(22
22
)sin(4
sin
)cos(4
sin4
cos)sin(4
sin
xxx
xxx
xxx
Simplify
using a sum or difference identity
2cos
x
)()sin()cos()cos()cos( bcinababa
In order to answer this question, we need to use the cosine formula for the difference of two angles.
continued on next slide
For the formula a will be 2
x and b will be
2sin)sin(
2cos)cos(
2cos
xxx
Simplify
using a sum or difference identity
2cos
x
)sin(2
cos
)1)(sin()0)(cos(2
cos
2sin)sin(
2cos)cos(
2cos
xx
xxx
xxx
Find the exact value of the following trigonometric functions below given
IVquadrantinisand73
cos
IIquadrantinisand54
sin
cos.1
sin.2
and
continued on next slide
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.
Find the exact value of the following trigonometric functions below given
IVquadrantinisand73
cos
IIquadrantinisand54
sin
and
continued on next slide
Triangle for α
b
3
α
7
40
positiveislength
40
40
499
73
2
2
222
b
b
b
b
b
Find the exact value of the following trigonometric functions below given
IVquadrantinisand73
cos
IIquadrantinisand54
sin
and
continued on next slide
Triangle for β
4
a
β
5
3
positiveislength
9
9
2516
54
2
2
222
a
a
a
a
a
Find the exact value of the following trigonometric functions below given
IVquadrantinisand73
cos
IIquadrantinisand54
sin
cos.1
and
continued on next slide
3
α
7
4
3
β
5
40
Now that we have our triangles, we can use the cosine identity for the sum of two angles to complete the problem.
54
740
53
73
)cos(
)sin()sin()cos()cos()cos(
aa
Find the exact value of the following trigonometric functions below given
IVquadrantinisand73
cos
IIquadrantinisand54
sin
cos.1
and
continued on next slide
3
α
7
4
3
β
5
40
Now that we have our triangles, we can use the cosine identity for the sum of two angles to complete the problem.
54
740
53
73
)cos(
)sin()sin()cos()cos()cos(
aa Note: Since β is in quadrant II, the cosine value will be negative
Note: Since α is in quadrant Iv, the sine value will be negative
Find the exact value of the following trigonometric functions below given
IVquadrantinisand73
cos
IIquadrantinisand54
sin
cos.1
and
continued on next slide
3
α
7
4
3
β
5
40
354049
)cos(
35404
359
)cos(
54
740
53
73
)cos(
Find the exact value of the following trigonometric functions below given
IVquadrantinisand73
cos
IIquadrantinisand54
sin
cos.1
and
continued on next slide
354049
)cos(
While we are here, what are the possible quadrants in which the angle α+β can fall?
In order to answer this question, we need to know if cos(α+β) is positive or negative. We can type the value into the calculator to determine this. When we do this, we find that cos(α+β) is positive. The cosine if positive in quadrants I and IV. Thus α+β must be in either quadrant I or IV. We cannot narrow our answer down any further without knowing the sign of sin(α+β).
Find the exact value of the following trigonometric functions below given
IVquadrantinisand73
cos
IIquadrantinisand54
sin
sin.2
and
continued on next slide
3
α
7
4
3
β
5
40
Now that we have our triangles, we can use the cosine identity for the sum of two angles to complete the problem.
54
73
53
740
)sin(
)sin()cos()cos()sin()sin(
aa Note: Since β is in quadrant II, the cosine value will be negative
Note: Since α is in quadrant Iv, the sine value will be negative
Find the exact value of the following trigonometric functions below given
IVquadrantinisand73
cos
IIquadrantinisand54
sin
sin.2
and
3
α
7
4
3
β
5
40
3512403
)sin(
3512
35403
)sin(
54
73
53
740
)sin(
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