Comprehensive Slideshow for Precalculus with Analytic and Trigonometry Concepts

In this slideshow accompanying the Precalculus textbook by Richard Wright, diverse topics such as Analytic, Trigonometry, and Precalculus Chapter 05 are discussed with examples and diagrams from the book. The content is designed to assist learners in gaining a better understanding of key mathematical concepts essential for further studies in the field.

• Precalculus
• Analytic
• Trigonometry
• Mathematics
• Education

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1. Analytic Analytic Trigonometry Trigonometry Precalculus Chapter 05

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 5- -01 Fundamental 01 Fundamental Trigonometric Trigonometric Identities Part A Identities Part A In this section, you will: Use fundamental identities to evaluate trigonometric expressions. Use fundamental identities to simplify trigonometric expressions.

4. 5 5- -01 Fundamental Trigonometric 01 Fundamental Trigonometric Identities Part A Identities Part A Uses for identities Evaluate trig functions Simplify trig expressions Develop more identities Solve trig equations

5. 5 5- -01 Fundamental Trigonometric 01 Fundamental Trigonometric Identities Part A Identities Part A Reciprocal Identities sin? = csc? = sin ? cos? = sec? = cos ? tan? = cot? = tan ? Pythagorean Identites sin2? + cos2? = 1 tan2? + 1 = sec2? 1 + cot2? = csc2? 1 1 csc ? 1 sec ? 1 cot ? 1 1 Quotient Identities tan? = sin ? cos ? cot? =cos ? sin ?

6. 5 5- -01 Fundamental Trigonometric 01 Fundamental Trigonometric Identities Part A Identities Part A Even/Odd Identities cos ? = cos? sec ? = sec? sin ? = sin? tan ? = tan? csc ? = csc? cot ? = cot? Cofunction Identities ? 2 ? = cos? cos 2 ? = sin? tan 2 ? = cot? cot 2 ? = tan? sec 2 ? = csc? csc 2 ? = sec? sin ? ? ? ? ?

7. 5 5- -01 Fundamental Trigonometric 01 Fundamental Trigonometric Identities Part A Identities Part A If sin? = 1 and cot? = 0, evaluate cos? Evaluate tan?

8. 5 5- -01 Fundamental Trigonometric 01 Fundamental Trigonometric Identities Part A Identities Part A Simplify sec2? 1 sin2?

9. 5 5- -01 Fundamental Trigonometric 01 Fundamental Trigonometric Identities Part A Identities Part A Simplify sin csc? sin?

10. 5 5- -01 Fundamental Trigonometric 01 Fundamental Trigonometric Identities Part A Identities Part A Simplify 1 sin2? csc2? 1

11. 5 5- -01 Fundamental Trigonometric 01 Fundamental Trigonometric Identities Part A Identities Part A Simplify cos 2 ? sec? ?

12. 5 5- -02 Fundamental 02 Fundamental Trigonometric Trigonometric Identities Part B Identities Part B In this section, you will: Factor and multiply trigonometric expressions. Use trigonometric identities with rational expressions. Use trigonometric substitution.

13. 5 5- -02 Fundamental Trigonometric 02 Fundamental Trigonometric Identities Part B Identities Part B Factor and simplify sin4? cos4?

14. 5 5- -02 Fundamental Trigonometric 02 Fundamental Trigonometric Identities Part B Identities Part B Multiply and simplify 2csc? + 2 2csc? 2

15. 5 5- -02 Fundamental Trigonometric 02 Fundamental Trigonometric Identities Part B Identities Part B cos ? 1+sin ?+1+sin ? cos ? Simplify

16. 5 5- -02 Fundamental Trigonometric 02 Fundamental Trigonometric Identities Part B Identities Part B Rewrite not as a fraction: sec ? tan ? 3

17. 5 5- -02 Fundamental Trigonometric 02 Fundamental Trigonometric Identities Part B Identities Part B Use trig substitution: ?2 9 with ? = 3sec?

18. 5 5- -03 Verify 03 Verify Trigonometric Trigonometric Identities Identities In this section, you will: Verify trigonometric identities algebraically. Verify trigonometric identities graphically.

19. 5 5- -03 Verify Trigonometric Identities 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 5- -03 Verify Trigonometric Identities 03 Verify Trigonometric Identities Verify 1 + sin? 1 sin? = cos2?

21. 5 5- -03 Verify Trigonometric Identities 03 Verify Trigonometric Identities Verify sin2? sin4? = cos2? cos4?

22. 5 5- -03 Verify Trigonometric Identities 03 Verify Trigonometric Identities Verify cot2? csc ?= csc? sin?

23. 5 5- -03 Verify Trigonometric Identities 03 Verify Trigonometric Identities 1 Verify sec ? tan ?= csc? sin?

24. 5 5- -03 Verify Trigonometric Identities 03 Verify Trigonometric Identities Verify cos ? cot ? 1 = csc? 1 sin ?

25. 5 5- -04 Solve Trigonometric 04 Solve Trigonometric Equations 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 trigonometric equations.

26. 5 5- -04 Solve Trigonometric Equations 04 Solve Trigonometric Equations Main goal Isolate a trig expression Try identities to simplify Try solving by factoring Number of solutions sin? = 0 Infinite solutions so describe 0 + ?? = ??

27. 5 5- -04 Solve Trigonometric Equations 04 Solve Trigonometric Equations Solve sin? 2 = sin?

28. 5 5- -04 Solve Trigonometric Equations 04 Solve Trigonometric Equations Solve 4sin2? 3 = 0

29. 5 5- -04 Solve Trigonometric Equations 04 Solve Trigonometric Equations Solve sin2? = 2sin?

30. 5 5- -04 Solve Trigonometric Equations 04 Solve Trigonometric Equations Solve 3sec2? 2tan2? 4 = 0

31. 5 5- -04 Solve Trigonometric Equations 04 Solve Trigonometric Equations Solve in the interval [0, 2 ) sin? + 1 = cos?

32. 5 5- -04 Solve Trigonometric Equations 04 Solve Trigonometric Equations Solve on the interval [0, 2 ) 3 sin2? = 2

33. 5 5- -04 Solve Trigonometric Equations 04 Solve Trigonometric Equations Solve 4tan2? + 5tan? = 6

34. 5 5- -05 Sum and Difference 05 Sum and Difference Formulas 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 5- -05 Sum and Difference Formulas 05 Sum and Difference Formulas sin ? + ? = sin?cos? + cos?sin? sin ? ? = sin?cos? cos?sin? cos ? + ? = cos?cos? sin?sin? cos ? ? = cos?cos? + sin?sin? tan ?+tan ? 1 tan ? tan ? tan ? tan ? 1+tan ? tan ? tan ? + ? = tan ? ? =

36. 5 5- -05 Sum and Difference Formulas 05 Sum and Difference Formulas Use a sum or difference formula to find the exact value of tan255

37. 5 5- -05 Sum and Difference Formulas 05 Sum and Difference Formulas Find the exact value of cos95 cos35 + sin95 sin35

38. 5 5- -05 Sum and Difference Formulas 05 Sum and Difference Formulas Derive a reduction formula for sin ? +? 2

39. 5 5- -05 Sum and Difference Formulas 05 Sum and Difference Formulas Find all solutions in [0, 2 ) cos ? ? 3 + cos ? +? = 1 3

40. 5 5- -06 Multiple Angle 06 Multiple Angle Formulas 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 5- -06 Multiple Angle Formulas 06 Multiple Angle Formulas Double-Angle Formulas sin2? = 2sin?cos? cos2? = cos2? sin2? = 2cos2? 1 = 1 2sin2? tan2? = 1 tan2? 2 tan ?

42. 5 5- -06 Multiple Angle Formulas 06 Multiple Angle Formulas cos2? If sin? =3 Find sin2? 5 and 0 < ? <? 2, tan2?

43. 5 5- -06 Multiple Angle Formulas 06 Multiple Angle Formulas Derive a triple angle formula for cos3?

44. 5 5- -06 Multiple Angle Formulas 06 Multiple Angle Formulas Power-Reducing Formulas sin2? =1 cos 2? 2 cos2? =1+cos 2? 2 tan2? =1 cos 2? 1+cos 2?

45. 5 5- -06 Multiple Angle Formulas 06 Multiple Angle Formulas Rewrite cos4? as a sum of 1st powers of cosines.

46. 5 5- -06 Multiple Angle Formulas 06 Multiple Angle Formulas Half-Angle Formulas Find the exact value of cos105 sin? 1 cos ? 2 2= cos? 1+cos ? 2 2= 2=1 cos ? = 1+cos ? tan? sin ? sin ?

47. 5 5- -07 Product 07 Product- -to Formulas 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. to- -Sum Sum

48. 5 5- -07 Product 07 Product- -to to- -Sum Formulas Sum Formulas Product-to-Sum Formulas sin?sin? =1 cos?cos? =1 sin?cos? =1 cos?sin? =1 2cos ? ? cos ? + ? 2cos ? ? + cos ? + ? 2sin ? + ? + sin ? ? 2sin ? + ? sin ? ?

49. 5 5- -07 Product 07 Product- -to to- -Sum Formulas Sum Formulas Rewrite sin5?cos3? as a sum or difference.

50. 5 5- -07 Product 07 Product- -to to- -Sum Formulas Sum Formulas Sum-to-Product Formulas ?+? 2 ?+? 2 ?+? ? ? 2 ? ? 2 ? ? sin? + sin? = 2sin cos sin? sin? = 2cos sin cos? + cos? = 2cos cos 2 ?+? 2 2 ? ? 2 cos? cos? = 2sin sin

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