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(