1. 7.2 Addition & Subtraction Formulas
Revelation 22:17
The Spirit and the bride say, "Come!" And let him who
hears say, "Come!" Whoever is thirsty, let him come;
and whoever wishes, let him take the free gift of the
water of life.

2. Addition & Subtraction Formulas
These are on your help sheet

3. Addition & Subtraction Formulas
These are on your help sheet
Take a look at these ... but why do we have them?

4. Addition & Subtraction Formulas
These are on your help sheet
Take a look at these ... but why do we have them?
The Unit Circle allows us to find the trig values for
“pretty points” ... like sin(30°) or cos(135°).
To get exact trig values for non-pretty angles, we
can use the Addition & Subtraction identities.

5. Addition & Subtraction Formulas
These are on your help sheet
Take a look at these ... but why do we have them?
The Unit Circle allows us to find the trig values for
“pretty points” ... like sin(30°) or cos(135°).
To get exact trig values for non-pretty angles, we
can use the Addition & Subtraction identities.
Examples:
sin ( 75° ) is sin ( 45° + 30° )
cos ( 265° ) is cos ( 310° − 45° )

6. Find the exact value of sin(105°)

7. Find the exact value of sin(105°)
We need to find 2 “magic point” angles on the Unit
Circle that either have a sum or a difference to
equal 105 degrees.

8. Find the exact value of sin(105°)
We need to find 2 “magic point” angles on the Unit
Circle that either have a sum or a difference to
equal 105 degrees.
105° = 60° + 45°

9. Find the exact value of sin(105°)
We need to find 2 “magic point” angles on the Unit
Circle that either have a sum or a difference to
equal 105 degrees.
105° = 60° + 45°
sin (105° ) = sin ( 60° + 45° )

10. Find the exact value of sin(105°)
We need to find 2 “magic point” angles on the Unit
Circle that either have a sum or a difference to
equal 105 degrees.
105° = 60° + 45°
sin (105° ) = sin ( 60° + 45° )
= sin 60°cos 45° + cos 60°sin 45°

11. Find the exact value of sin(105°)
We need to find 2 “magic point” angles on the Unit
Circle that either have a sum or a difference to
equal 105 degrees.
105° = 60° + 45°
sin (105° ) = sin ( 60° + 45° )
= sin 60°cos 45° + cos 60°sin 45°
3 2 1 2
= ⋅ + ⋅
2 2 2 2

12. Find the exact value of sin(105°)
We need to find 2 “magic point” angles on the Unit
Circle that either have a sum or a difference to
equal 105 degrees.
105° = 60° + 45°
sin (105° ) = sin ( 60° + 45° )
= sin 60°cos 45° + cos 60°sin 45°
3 2 1 2
= ⋅ + ⋅
2 2 2 2
6+ 2
=
4

13. ⎛ 5π ⎞
Find the exact value of cos ⎜ ⎟
⎝ 12 ⎠

14. ⎛ 5π ⎞
Find the exact value of cos ⎜ ⎟
⎝ 12 ⎠
8π 3π
−
12 12
2π π
−
3 4

15. ⎛ 5π ⎞
Find the exact value of cos ⎜ ⎟
⎝ 12 ⎠
8π 3π
−
12 12
2π π
or
−
3 4

16. ⎛ 5π ⎞
Find the exact value of cos ⎜ ⎟
⎝ 12 ⎠
8π 3π 2π 3π
− +
12 12 12 12
2π π
or π π
− +
3 4 6 4

17. ⎛ 5π ⎞
Find the exact value of cos ⎜ ⎟
⎝ 12 ⎠
8π 3π 2π 3π
− +
12 12 12 12
2π π
or π π
− +
3 4 6 4
⎛ 5π ⎞ ⎛ 2π π ⎞
cos ⎜ ⎟ = cos ⎜ − ⎟
⎝ 12 ⎠ ⎝ 3 4 ⎠

18. ⎛ 5π ⎞
Find the exact value of cos ⎜ ⎟
⎝ 12 ⎠
8π 3π 2π 3π
− +
12 12 12 12
2π π
or π π
− +
3 4 6 4
⎛ 5π ⎞ ⎛ 2π π ⎞
cos ⎜ ⎟ = cos ⎜ − ⎟
⎝ 12 ⎠ ⎝ 3 4 ⎠
2π π 2π π
= cos cos + sin sin
3 4 3 4

19. ⎛ 5π ⎞
Find the exact value of cos ⎜ ⎟
⎝ 12 ⎠
8π 3π 2π 3π
− +
12 12 12 12
2π π
or π π
− +
3 4 6 4
⎛ 5π ⎞ ⎛ 2π π ⎞
cos ⎜ ⎟ = cos ⎜ − ⎟
⎝ 12 ⎠ ⎝ 3 4 ⎠
2π π 2π π
= cos cos + sin sin
3 4 3 4
⎛ 1 ⎞ ⎛ 2 ⎞ ⎛ 3 ⎞ ⎛ 2 ⎞
= ⎜ − ⎟ ⎜ ⎟ + ⎜ 2 ⎟ ⎜ 2 ⎟
⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ ⎠ ⎝ ⎠

20. ⎛ 5π ⎞
Find the exact value of cos ⎜ ⎟
⎝ 12 ⎠
8π 3π 2π 3π
− +
12 12 12 12
2π π
or π π
− +
3 4 6 4
⎛ 5π ⎞ ⎛ 2π π ⎞
cos ⎜ ⎟ = cos ⎜ − ⎟
⎝ 12 ⎠ ⎝ 3 4 ⎠ 2 6
=− +
4 4
2π π 2π π
= cos cos + sin sin
3 4 3 4
⎛ 1 ⎞ ⎛ 2 ⎞ ⎛ 3 ⎞ ⎛ 2 ⎞
= ⎜ − ⎟ ⎜ ⎟ + ⎜ 2 ⎟ ⎜ 2 ⎟
⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ ⎠ ⎝ ⎠

21. ⎛ 5π ⎞
Find the exact value of cos ⎜ ⎟
⎝ 12 ⎠
8π 3π 2π 3π
− +
12 12 12 12
2π π
or π π
− +
3 4 6 4
⎛ 5π ⎞ ⎛ 2π π ⎞
cos ⎜ ⎟ = cos ⎜ − ⎟
⎝ 12 ⎠ ⎝ 3 4 ⎠ 2 6
=− +
4 4
2π π 2π π
= cos cos + sin sin
3 4 3 4
6− 2
=
⎛ 1 ⎞ ⎛ 2 ⎞ ⎛ 3 ⎞ ⎛ 2 ⎞ 4
= ⎜ − ⎟ ⎜ ⎟ + ⎜ 2 ⎟ ⎜ 2 ⎟
⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ ⎠ ⎝ ⎠

22. Prove the identity:

23. Prove the identity:
⎛ π ⎞ 2 ( cos α + sin α )
cos ⎜ α − ⎟ =
⎝ 4 ⎠ 2

24. Prove the identity:
⎛ π ⎞ 2 ( cos α + sin α )
cos ⎜ α − ⎟ =
⎝ 4 ⎠ 2
π π
cos α cos + sin α sin =
4 4

25. Prove the identity:
⎛ π ⎞ 2 ( cos α + sin α )
cos ⎜ α − ⎟ =
⎝ 4 ⎠ 2
π π
cos α cos + sin α sin =
4 4
2 2
cos α + sin α =
2 2

26. Prove the identity:
⎛ π ⎞ 2 ( cos α + sin α )
cos ⎜ α − ⎟ =
⎝ 4 ⎠ 2
π π
cos α cos + sin α sin =
4 4
2 2
cos α + sin α =
2 2
2
( cos α + sin α ) =
2

27. Prove the identity:

28. Prove the identity:
⎛ π ⎞ tan θ + 1
tan ⎜ θ + ⎟ =
⎝ 4 ⎠ 1− tan θ

29. Prove the identity:
⎛ π ⎞ tan θ + 1
tan ⎜ θ + ⎟ =
⎝ 4 ⎠ 1− tan θ
π
tan θ + tan
4 =
π
1− tan θ tan
4

30. Prove the identity:
⎛ π ⎞ tan θ + 1
tan ⎜ θ + ⎟ =
⎝ 4 ⎠ 1− tan θ
π
tan θ + tan
4 =
π
1− tan θ tan
4
tan θ + 1
=
1− tan θ (1)

31. Prove the identity:
⎛ π ⎞ tan θ + 1
tan ⎜ θ + ⎟ =
⎝ 4 ⎠ 1− tan θ
π
tan θ + tan
4 =
π
1− tan θ tan
4
tan θ + 1
=
1− tan θ (1)
tan θ + 1
=
1− tan θ

32. We are skipping the part entitled,
Expressions of the Form Asinx + Bcosx

33. We are skipping the part entitled,
Expressions of the Form Asinx + Bcosx
If time allows, let’s derive the
Difference Identity for Cosine

34. We are skipping the part entitled,
Expressions of the Form Asinx + Bcosx
If time allows, let’s derive the
Difference Identity for Cosine
Our procedure will be:

35. We are skipping the part entitled,
Expressions of the Form Asinx + Bcosx
If time allows, let’s derive the
Difference Identity for Cosine
Our procedure will be:
1. use law of cosines to find AB

36. We are skipping the part entitled,
Expressions of the Form Asinx + Bcosx
If time allows, let’s derive the
Difference Identity for Cosine
Our procedure will be:
1. use law of cosines to find AB
2. use distance formula to find AB

37. We are skipping the part entitled,
Expressions of the Form Asinx + Bcosx
If time allows, let’s derive the
Difference Identity for Cosine
Our procedure will be:
1. use law of cosines to find AB
2. use distance formula to find AB
3. equate these two results and solve

38. cos (α − β ) = cos α cos β + sin α sin β

39. cos (α − β ) = cos α cos β + sin α sin β
1. Use Law of Cosines to find AB

40. cos (α − β ) = cos α cos β + sin α sin β
1. Use Law of Cosines to find AB
2 2 2
(AB) = 1 + 1 − 2(1)(1)cos (α − β )

41. cos (α − β ) = cos α cos β + sin α sin β
1. Use Law of Cosines to find AB
2 2 2
(AB) = 1 + 1 − 2(1)(1)cos (α − β )
2
(AB) = 2 − 2 cos (α − β )

42. cos (α − β ) = cos α cos β + sin α sin β
1. Use Law of Cosines to find AB
2 2 2
(AB) = 1 + 1 − 2(1)(1)cos (α − β )
2
(AB) = 2 − 2 cos (α − β )
AB = 2 − 2 cos (α − β )

43. 2. Use Distance Formula to find AB

44. 2. Use Distance Formula to find AB
2 2
AB = ( cos α − cos β ) + (sin α − sin β )

45. 2. Use Distance Formula to find AB
2 2
AB = ( cos α − cos β ) + (sin α − sin β )
2 2 2 2
AB = cos α − 2 cos α cos β + cos β + sin α − 2sin α sin β + sin β

46. 2. Use Distance Formula to find AB
2 2
AB = ( cos α − cos β ) + (sin α − sin β )
2 2 2 2
AB = cos α − 2 cos α cos β + cos β + sin α − 2sin α sin β + sin β
AB = (sin α + cos α ) + (sin
2 2 2
β + cos β ) − 2 cos α cos β − 2sin α sin β
2

47. 2. Use Distance Formula to find AB
2 2
AB = ( cos α − cos β ) + (sin α − sin β )
2 2 2 2
AB = cos α − 2 cos α cos β + cos β + sin α − 2sin α sin β + sin β
AB = (sin α + cos α ) + (sin
2 2 2
β + cos β ) − 2 cos α cos β − 2sin α sin β
2
AB = 2 − 2 cos α cos β − 2sin α sin β

48. 3. Equate the two and solve

49. 3. Equate the two and solve
2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β

50. 3. Equate the two and solve
2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β
2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β

51. 3. Equate the two and solve
2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β
2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β
−2 cos (α − β ) = −2 cos α cos β − 2sin α sin β

52. 3. Equate the two and solve
2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β
2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β
−2 cos (α − β ) = −2 cos α cos β − 2sin α sin β
−2 cos (α − β ) = −2 ( cos α cos β + sin α sin β )

53. 3. Equate the two and solve
2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β
2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β
−2 cos (α − β ) = −2 cos α cos β − 2sin α sin β
−2 cos (α − β ) = −2 ( cos α cos β + sin α sin β )
cos (α − β ) = cos α cos β + sin α sin β

54. 3. Equate the two and solve
2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β
2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β
−2 cos (α − β ) = −2 cos α cos β − 2sin α sin β
−2 cos (α − β ) = −2 ( cos α cos β + sin α sin β )
cos (α − β ) = cos α cos β + sin α sin β
And this is the Difference Identity for Cosine
We could derive all the others in a similar fashion ...

55. HW #4
Today is just a good day in disguise.
Paul Venghaus