Precalculus 05 Analytic Trigonometry (1).pptx
- 2. ▴This Slideshow was developed to accompany the textbook ▸Precalculus ▸By Richard Wright ▸https://www.andrews.edu/~rwright/Precalculus-RLW/Text/ TOC.html ▴Some examples and diagrams are taken from the textbook. Slides created by Richard Wright, Andrews Academy rwright@andrews.edu
- 3. 5-01 Fundamental Trigonometric Identities Part A In this section, you will: • Use fundamental identities to evaluate trigonometric expressions. • Use fundamental identities to simplify trigonometric expressions.
- 4. 5-01 Fundamental Trigonometric Identities Part A ▴Uses for identities ▸Evaluate trig functions ▸Simplify trig expressions ▸Develop more identities ▸Solve trig equations
- 5. 5-01 Fundamental Trigonometric Identities Part A ▴Reciprocal Identities ▴ ▴ ▴ ▴Quotient Identities ▴ ▴Pythagorean Identites
- 6. 5-01 Fundamental Trigonometric Identities Part A ▴Even/Odd Identities ▴ ▴Cofunction Identities
- 7. 5-01 Fundamental Trigonometric Identities Part A ▴If and , evaluate ▴Evaluate
- 8. 5-01 Fundamental Trigonometric Identities Part A ▴Simplify
- 9. 5-01 Fundamental Trigonometric Identities Part A ▴Simplify
- 10. 5-01 Fundamental Trigonometric Identities Part A ▴Simplify
- 11. 5-01 Fundamental Trigonometric Identities Part A ▴Simplify
- 12. 5-02 Fundamental Trigonometric Identities Part B In this section, you will: • Factor and multiply trigonometric expressions. • Use trigonometric identities with rational expressions. • Use trigonometric substitution.
- 13. 5-02 Fundamental Trigonometric Identities Part B ▴Factor and simplify
- 14. 5-02 Fundamental Trigonometric Identities Part B ▴Multiply and simplify
- 15. 5-02 Fundamental Trigonometric Identities Part B ▴Simplify
- 16. 5-02 Fundamental Trigonometric Identities Part B ▴Rewrite not as a fraction:
- 17. 5-02 Fundamental Trigonometric Identities Part B ▴Use trig substitution: with
- 18. 5-03 Verify Trigonometric Identities In this section, you will: • Verify trigonometric identities algebraically. • Verify trigonometric identities graphically.
- 19. 5-03 Verify Trigonometric Identities ▴Show that trig identities are true by turning one side into the other side ▴Guidelines 1. Work with 1 side at a time. Start with the more complicated side. 2. Try factor, add fractions, square a binomial, etc. 3. Use fundamental identities 4. If the above doesn’t work, try rewriting in sines and cosines 5. Try something!
- 20. 5-03 Verify Trigonometric Identities ▴Verify
- 21. 5-03 Verify Trigonometric Identities ▴Verify
- 22. 5-03 Verify Trigonometric Identities ▴Verify
- 23. 5-03 Verify Trigonometric Identities ▴Verify
- 24. 5-03 Verify Trigonometric Identities ▴Verify
- 25. 5-04 Solve Trigonometric Equations In this section, you will: • Solve trigonometric simple equations. • Solve trigonometric equations by factoring. • Solve trigonometric equations using identities. • Find all solutions and all solutions on the interval [0, 2 ) to π trigonometric equations.
- 26. 5-04 Solve Trigonometric Equations ▴Main goal – Isolate a trig expression ▸Try identities to simplify ▸Try solving by factoring ▴Number of solutions ▸Infinite solutions so describe
- 27. 5-04 Solve Trigonometric Equations ▴Solve
- 28. 5-04 Solve Trigonometric Equations ▴Solve
- 29. 5-04 Solve Trigonometric Equations ▴Solve
- 30. 5-04 Solve Trigonometric Equations ▴Solve
- 31. 5-04 Solve Trigonometric Equations ▴Solve in the interval [0, 2π)
- 32. 5-04 Solve Trigonometric Equations ▴Solve on the interval [0, 2π)
- 33. 5-04 Solve Trigonometric Equations ▴Solve
- 34. 5-05 Sum and Difference Formulas In this section, you will: • Apply the sum and difference formulas to evaluate trigonometric expressions. • Apply the sum and difference formulas to simplify trigonometric expressions. • Apply the sum and difference formulas to solve trigonometric equations.
- 35. 5-05 Sum and Difference Formulas
- 36. 5-05 Sum and Difference Formulas ▴Use a sum or difference formula to find the exact value of
- 37. 5-05 Sum and Difference Formulas ▴Find the exact value of
- 38. 5-05 Sum and Difference Formulas ▴Derive a reduction formula for
- 39. 5-05 Sum and Difference Formulas ▴Find all solutions in [0, 2π)
- 40. 5-06 Multiple Angle Formulas In this section, you will: • Use multiple angle formulas to evaluate trigonometric functions. • Use multiple angle formulas to derive new trigonometric identities. • Use multiple angle formulas to solve trigonometric equations.
- 41. 5-06 Multiple Angle Formulas ▴Double-Angle Formulas
- 42. 5-06 Multiple Angle Formulas ▴If and , ▴Find
- 43. 5-06 Multiple Angle Formulas ▴Derive a triple angle formula for
- 44. 5-06 Multiple Angle Formulas ▴Power-Reducing Formulas
- 45. 5-06 Multiple Angle Formulas ▴Rewrite as a sum of 1st powers of cosines.
- 46. 5-06 Multiple Angle Formulas ▴Half-Angle Formulas ▴Find the exact value of
- 47. 5-07 Product-to-Sum Formulas In this section, you will: • Use product-to-sum formulas to evaluate trigonometric functions. • Use product-to-sum formulas to derive new trigonometric identities. • Use product-to-sum formulas to solve trigonometric equations.
- 48. 5-07 Product-to-Sum Formulas ▴Product-to-Sum Formulas
- 49. 5-07 Product-to-Sum Formulas ▴Rewrite as a sum or difference.
- 50. 5-07 Product-to-Sum Formulas ▴Sum-to-Product Formulas
- 51. 5-07 Product-to-Sum Formulas ▴Find the exact value of
- 52. 5-07 Product-to-Sum Formulas ▴Solve on the interval [0, 2π)
Editor's Notes
- #5: Colored ones should be memorized.
- #27: Use a unit circle to find solutions
- #28: Use a unit circle to find the solutions It is +πn because solutions are directly opposite each other on the circle so that adding π moves to another solution.
- #29: Use a unit circle for each equation to find the solutions Only solution is
- #30: Use a unit circle to find all the soutions
- #31: Use a unit circle for each equation to find all solutions between 0 and 2π
- #32: Use a unit circle to find all the solutions. Because 2x, you need to go around circle twice
- #33: Quadratic type Use a unit circle for each equation
- #41: These come from the sum formulas where u = v
- #42: Use a right triangle in the first quadrant with y = 3 and r = 5 to find x = 4
- #46: Pick the negative because 105° falls in quadrant II where cos is negative
- #52: Use a unit circle for each equation. The cos 3x needs to be gone around 3 times