1. 4.3 – Trigonometric Functions on the Unit Circle

2. Trigonometric Functions of Any Angle Given the diagram, r = √x 2 + y 2 by the Pythagorean Theorem and the trigonometric functions are as follows:

3. Trigonometric Functions of Any Angle Given the diagram, r = √x 2 + y 2 by the Pythagorean Theorem and the trigonometric functions are as follows: sinƟ = y r

4. Trigonometric Functions of Any Angle Given the diagram, r = √x 2 + y 2 by the Pythagorean Theorem and the trigonometric functions are as follows: sinƟ = ycscƟ = r r y

5. Trigonometric Functions of Any Angle Given the diagram, r = √x 2 + y 2 by the Pythagorean Theorem and the trigonometric functions are as follows: sinƟ = ycscƟ = r r y cosƟ = x r

6. Trigonometric Functions of Any Angle Given the diagram, r = √x 2 + y 2 by the Pythagorean Theorem and the trigonometric functions are as follows: sinƟ = ycscƟ = r r y cosƟ = xsecƟ = r r x

7. Trigonometric Functions of Any Angle Given the diagram, r = √x 2 + y 2 by the Pythagorean Theorem and the trigonometric functions are as follows: sinƟ = ycscƟ = r r y cosƟ = xsecƟ = r r x tanƟ = y x

8. Trigonometric Functions of Any Angle Given the diagram, r = √x 2 + y 2 by the Pythagorean Theorem and the trigonometric functions are as follows: sinƟ = ycscƟ = r r y cosƟ = xsecƟ = r r x tanƟ = ycotƟ = x x y

9. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ.

10. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2

11. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2

12. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10

13. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 10

14. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 = -3 10 5

15. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 = -3cscƟ = 10 10 5 -6

16. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 = -3cscƟ = 10 = -5 10 5 -6 3

17. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 = -3cscƟ = 10 = -5 10 5 -6 3 cosƟ = 8 10

18. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 = -3cscƟ = 10 = -5 10 5 -6 3 cosƟ = 8 = 4 10 5

19. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 = -3cscƟ = 10 = -5 10 5 -6 3 cosƟ = 8 = 4secƟ = 10 10 5 8

20. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 = -3cscƟ = 10 = -5 10 5 -6 3 cosƟ = 8 = 4secƟ = 10 = 5 10 5 8 4

21. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 = -3cscƟ = 10 = -5 10 5 -6 3 cosƟ = 8 = 4secƟ = 10 = 5 10 5 8 4 tanƟ = -6 8

22. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 = -3cscƟ = 10 = -5 10 5 -6 3 cosƟ = 8 = 4secƟ = 10 = 5 10 5 8 4 tanƟ = -6 = -3 8 4

23. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 = -3cscƟ = 10 = -5 10 5 -6 3 cosƟ = 8 = 4secƟ = 10 = 5 10 5 8 4 tanƟ = -6 = -3cotƟ = 8 8 4 -6

24. Ex. 1 Let (8,-6) be a point on the terminal side of an angle Ɵ in standard position. Find the exact values of the six trigonometric functions of Ɵ. r = √x 2 + y 2 r = √(8) 2 + (-6) 2 r = 10 sinƟ = -6 = -3cscƟ = 10 = -5 10 5 -6 3 cosƟ = 8 = 4secƟ = 10 = 5 10 5 8 4 tanƟ = -6 = -3cotƟ = 8 = -4 8 4 -6 3

25. If sinƟ = opp, cosƟ = adj, and tanƟ = opp, hyp hyp adj

26. If sinƟ = opp, cosƟ = adj, and tanƟ = opp, hyp hyp adj then sinƟ = opp / hyp cosƟ adj / hyp

27. If sinƟ = opp, cosƟ = adj, and tanƟ = opp, hyp hyp adj then sinƟ = opp / hyp = opp. hyp cosƟ adj / hyp hyp adj

28. If sinƟ = opp, cosƟ = adj, and tanƟ = opp, hyp hyp adj then sinƟ = opp / hyp = opp. hyp = opp cosƟ adj / hyp hyp adj adj

29. If sinƟ = opp, cosƟ = adj, and tanƟ = opp, hyp hyp adj then sinƟ = opp / hyp = opp. hyp = opp = tanƟ cosƟ adj / hyp hyp adj adj

30. If sinƟ = opp, cosƟ = adj, and tanƟ = opp, hyp hyp adj then sinƟ = opp / hyp = opp. hyp = opp = tanƟ cosƟ adj / hyp hyp adj adj So tanƟ = sinƟ cosƟ

31. Ex. 2 Find the exact value of each expression. a. cos210°

32. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2

33. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2 b. sin- 7π / 4

34. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2 b. sin- 7π / 4 = √2 2

35. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2 b. sin- 7π / 4 = √2 2 c. cos 11π / 4

36. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2 b. sin- 7π / 4 = √2 2 c. cos 11π / 4 = cos(2π + 3π / 4 )

37. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2 b. sin- 7π / 4 = √2 2 c. cos 11π / 4 = cos(2π + 3π / 4 ) = cos 3π / 4

38. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2 b. sin- 7π / 4 = √2 2 c. cos 11π / 4 = cos(2π + 3π / 4 ) = cos 3π / 4 = -√2 / 2

39. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2 b. sin- 7π / 4 = √2 2 c. cos 11π / 4 = cos(2π + 3π / 4 ) = cos 3π / 4 = -√2 / 2 d. tan π / 6

40. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2 b. sin- 7π / 4 = √2 2 c. cos 11π / 4 = cos(2π + 3π / 4 ) = cos 3π / 4 = -√2 / 2 d. tan π / 6 = sin π / 6 cos π / 6

41. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2 b. sin- 7π / 4 = √2 2 c. cos 11π / 4 = cos(2π + 3π / 4 ) = cos 3π / 4 = -√2 / 2 d. tan π / 6 = sin π / 6 = 1 / 2 cos π / 6 √3 / 2

42. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2 b. sin- 7π / 4 = √2 2 c. cos 11π / 4 = cos(2π + 3π / 4 ) = cos 3π / 4 = -√2 / 2 d. tan π / 6 = sin π / 6 = 1 / 2 = 1 cos π / 6 √3 / 2 √3

43. Ex. 2 Find the exact value of each expression. a. cos210° = -√3 2 b. sin- 7π / 4 = √2 2 c. cos 11π / 4 = cos(2π + 3π / 4 ) = cos 3π / 4 = -√2 / 2 d. tan π / 6 = sin π / 6 = 1 / 2 = 1 = √3 cos π / 6 √3 / 2 √3 3