1. Using Fundamental Trig Identities
Verifying Identities
And
Solving Trig Equations
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: December 29, 2009

2. Reciprocal Identities

3. Quotient Identities

4. Pythagorean Identities

5. Cofunction Identities
Complimentary Angles

6. Even/Odd Identities
EVEN
ODD

7. Use the given to evaluate all six trig functions
First determine that quadrant the given information holds true…….
What quadrant is cotangent negative and sine positive??? II
tangent

8. Simplify a Trig Expression

9. Verify a Trig Identity Work on one side only! Work DOWN the page,
not across!

10. Select the path style for y2 so you can see the tracing
Verify a Trig Identity
Use the table feature and graphing utility to check your result.
Select the path style for y2 so you can see the tracing

11. Verify a Trig Identity

12. Factoring Trig Expressions

13. Factoring Trig Expressions

14. Factoring Trig Expressions

15. Factoring Trig Expressions

16. Factoring Trig Expressions

17. Add & Subtract Trig Expressions

18. Add & Subtract Trig Expressions

19. Add & Subtract Trig Expressions

20. Verify Trig Identities

21. Verify Trig Identities - Guidelines
Work with one side of the equation at a time. It is often better to work with the more complicated side first.
Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator.
Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants with tangents, and cosecants with cotangents.
If the preceding guidelines do not help, try converting all terms to sines and cosines.
Always try something. Even making an attempt that leads to a dead end provides insight.

22. Verify Trig Identities

23. Verify Trig Identities

24. Verify Trig Identities

25. Verify Trig Identities

26. Verify Trig Identities

27. Verify Trig Identities

28. Verify Trig Identities

29. Verify Trig Identities

30. Verify Trig Identities

31. Verify Trig Identities

32. Solving Trig Equations

33. Solving Trig Equations

34. Solving Trig Equations

35. Solving Trig Equations

36. Solving Trig Equations

37. Solving Trig Equations

38. Solving Trig Equations

39. Sum & Difference Formulas

40. Sum & Difference Formulas

41. Proof of cos(A+B) = cosA•cosB - sinA•sinB
R (x2, y2 )  (cos(A+B), sin(A+B) )
Q (x1, y1)  (cosA, sinA) B
Q (x1, y1)
S (x3, y3) (cos(-B), sin(-B) )
R (x2, y2 ) A
P (1, 0) -B
S (x3, y3) 41

42. Proof of sin(A+B) = sinA•cosB + cosA•sinB
Note: This proof uses the cofunction identities.

43. Use Sum & Difference Formulas
Find the exact value of sin(750)

44. Use Sum & Difference Formulas
Find the exact value of:

45. Use Sum & Difference Formulas
Find the exact value of:
Rationalize the denominator

46. Use Sum & Difference Formulas
Simplfy:

47. Use Sum & Difference Formulas
Find the exact value of sin(u+v) given the following:

48. Verify Trig Identities

49. Solving Trig Equations on [0, 2π)

50. Double-Angle Formulas

51. Double-Angle Formulas

52. Double-Angle Formulas

53. Double-Angle Formulas
Re-cap
Double-Angle Formulas

54. Half-Angle Formulas

55. Solve for x

56. Use Double-Angle Formulas
Find the exact value of sin(2u), cos(2u) & tan(2u)

57. Use Half-Angle Formulas
Find the exact value of:

58. Use Half-Angle Formulas
Find the exact value of:

59. Solve for x

60. Product-to-Sum Formulas

61. Sum-to-Product Formulas

62. Power-Reducing Formulas

63. Use Power-Reducing Formulas
Rewrite sin4x in terms of the first power of cosine.