1. Trigonometric Identities
Simplifying

2. Some Vocab
Identity: a statement of equality between two expressions that is true for all values of the variable(s)
Trigonometric Identity: an identity involving trig expressions.

3. Reciprocal Identities

4. Quotient Identities
Know these!

5. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this:
substitute using each identity

6. substitute using each identity
Now, Simplify:

7. One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this:
substitute using each identity
simplify

8. Do you remember the Unit Circle?
What is the equation for the unit circle?
x2 + y2 = 1
What does x = ? What does y = ?
(in terms of trig functions)
sin2θ + cos2θ = 1
Pythagorean Identity!
ALSO USED AS: or

9. Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by cos2θ
sin2θ + cos2θ =
cos2θ cos2θ cos2θ
tan2θ = sec2θ
Quotient
Identity
Reciprocal
Identity
another Pythagorean Identity

10. Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by sin2θ
sin2θ + cos2θ =
sin2θ sin2θ sin2θ
cot2θ = csc2θ
Quotient
Identity
Reciprocal
Identity
a third Pythagorean Identity

11. RECIPROCAL IDENTITIES
QUOTIENT IDENTITIES
PYTHAGOREAN IDENTITIES
IF ONE WANTS A TAN, HE SEEKS THE SUN
IF ONE WANTS TO LOOK LIKE COTTON, HE COVERS WITH SUN SCREEN

12. Opposite Angle Identities
sometimes these are called even/odd identities

13. Simplify each expression.

14. Simplify each expression.

15. Simplify each expression.

16. Simplify the expressio .
Remember sin and cos:

17. Simplify with addition.
Here we need a
Common denominator!
Simplify here

18. Now: Prove the identity:
Work on the complicated side only

19. Now: Prove the identity:
Work on the complicated side only
Common denominator
Pythagorean identity

20. Substituting and factoring:
Ex:
This is # 41 on pg 452 precalc book

21. Substituting and factoring:
Ex:
This is # 41 on pg 452 precalc book
Think:

22. Substituting and factoring:
Ex:
This is # 41 on pg 452 precalc book

23. Substitute, distribute, simplify
Ex:
#43

24. Factor
Ex:
#43

25. Factor
Ex:
#43

26. Factoring to simplify:
Ex:
#47

27. Factoring to simplify:
Ex:
#47

28. Do Now:
Calculate: cos 60

29. Cos (A – B) From the reference sheet: COS (A-B)=cosAcosB + sinAsinB
Example:

30. COS (A – B) From the reference sheet: COS (A-B)=cosAcosB + sinAsinB
Example: =
= 0 + ½
= ½

31. Example
Find cos 15 exactly by using cos(45-30)

32. Example
Find cos 15 exactly by using cos(45-30)

33. Do Now: If
Hint: draw a triangle.

34. Do Now: If
If A and B are 2nd quadrant angles, find cos(A-B)

35. Do Now: If
If A and B are 2nd quadrant angles, find cos(A-B)

36. Sum and Difference Use sin (a+b)
To find the exact value of 75 degrees.

37. Sum and Difference Use sin (a+b)
To find the exact value of 75 degrees.

38. Remember the do now?
Use the same strategy for the next problem.

39. Finding the missing function first
Given that
Write out what you know so far…..
sinA cosB – cosA sinB

40. Finding the missing function first
Given that
sinA cosB – cosA sinB
Find the missing sides of both triangles 5 13 5 B A 4

41. Finding the missing function first
Given that
sinA cosB – cosA sinB 5 13 3 5 B A 4 12

42. Using the identities you now know, find the trig value.
If cosθ = 3/4, If cosθ = 3/5, find secθ. find cscθ.

43. sinθ = -1/3, 180o < θ < 270o; find tanθ secθ = -7/5, π < θ < 3π/2; find sinθ

44. Do Now:
Simplify:

45. Do Now:
Simplify:
Now Rationalize:

46. Do Now:
Rationalize:

47. Tan(A±B)
Use the reference sheet:
Find tan(45-30)

48. Tan(A±B)
Use the reference sheet:
Find tan(45-30)

49. Tan(A±B)
Use the reference sheet:
Find tan(45-30)

50. Tan(A±B)
Use the reference sheet:
Find tan(180+45)

51. Tan(A±B)
Use the reference sheet:
Find tan(180+45)

52. examples
Answer # 1 on page 502.

53. sin2A
Find sin2A if A =45
2sinAcosA=

54. sin2A
Find sin2A if A =45
2sinAcosA=
And the sin 90 = 1!

55. Cos A
Find cos 2A if A = 30

56. Cos A
Find cos 2A if A = 30

57. Which quadrant? If
Remember to use a triangle to find the cos. 3 5

58. Do Now: hint: Which quadrant?If

59. Do Now: hint: Which quadrant?If

60. sin ½ A, cos ½ A
See the reference sheet:
Find sin ½ A if A = 60

61. sin ½ A, cos ½ A
Find sin ½ A if A = 60 (must be positive – in quadrant I.

62. cos ½ A
Find cos ½ A if A = 60 (must be positive – in quadrant I.

63. cos ½ A
Find cos ½ A if A = 60 (must be positive – in quadrant I.

64. tan ½ A
Find tan ½ A if A = 90 (must be positive – in quadrant I.

65. tan ½ A
Find tan ½ A if A = 90 (must be positive – in quadrant I.

66. Which quadrant? If (quadrant III) Then (cos is negative in quad. III)
find 3 5 4
90< ½θ <135 which is in quadrant II
where sin is positive

67. Which quadrant? If
Or..

68. Blow pop question
Find sin(A+B) if
Hint: draw two triangles.

69. Blow pop question
Find sin(A+B) if