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.
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° )
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
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
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
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 )
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
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
Notas do Editor
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1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n