2. 0.2: Trignometric Functions Objective: Understand relationships between angles and sides of right triangles Review trig functions of special & quadrantal angles ©2008 Roy L. Gover (www.mrgover.com) Calculus Prerequisite Knowledge

3. Definition The special angles are: 30°,60°, and 45° in degrees and in radians In calculus, we use radians.

4. Important Idea These angles are special because they have exact value trig functions.

5. Consider the first two special angles… Long Side Short Side Hypotenuse

6. Long Side Hypotenuse Short Side …orientation does not change the relationships between sides and angles

7. Long Side Hypotenuse Short Side …orientation does not change the relationships between sides and angles

8. Analysis Consider an equilateral triangle with sides 2 units long 2 2 2

9. Analysis...equilateral triangles are equiangular and each angle must be 2 2 2

10. Analysis …the angle bisector intersects the base at the midpoint 2 2 11

11. Analysis 2 2 11 1 2 ?

12. Try This Find the height of an equilateral triangle with each side 4 inches long. 4 4 4 ?

13. Important Idea In a rad triangle: the short side is one-half the hypotenuse. the long side is times the short side. Memorize

14. Try This Find the length of the missing sides: 10 5

15. Try This Find the length of the missing sides: 5

16. Try This What is wrong with this picture?

17. Consider the last special angle…

18. Hypotenuse …orientation does not change the relationships between sides and angles

19. Hypotenuse …orientation does not change the relationships between sides and angles

20. …and sides opposite equal angles are equal... x x and by the pythagorean theorem, the hypotenuse is...

21. Important Idea In a triangle: The legs of the triangle are equal. the hypotenuse is times the length of the leg. Memorize

22. Try This Find the length of the missing sides 2 2

23. Try This Find the length of the missing sides 2

24. Review x y r  Memorize

25. Review x y r  Memorize

26. Find the exact values of the six trig Example functions of the given special angle in standard position.

27. Analysis 1. Make a right triangle. 

28. Analysis 2. Let the short side be 1, then use your knowledge of special angles 2 1 to find the measures of each side of the right triangle. 

29. Analysis 2 1 3. Use the definitions of the trig functions to find each trig function exact value. 

30. Analysis 2 1 

31. 2 1 

32. Find the exact values of the six trig Try This functions of the given special angle in standard position.

33. Solution

34. Important Idea If the given angle is other than a quadrant I angle, the exact values of the trig functions are calculated using the reference angle of the given angle.

35. Review Reference Angle: the angle between a given angle and the nearest x axis. (Note: x axis; not y axis). Reference angles are always positive.

36. Review How you find the reference angle depends on which quadrant contains the given angle.

37. Review

41. Example Draw a diagram showing the relevant angles and sides and find the sin, cos & tan of Given Angle Ref. Angle

42. Example Short Side Long Side Given Angle

43. Example 2 1 Why is 1 & 2 pos. & neg.?

44. Try This Draw a diagram showing the relevant angles and sides and find the csc, sec & cot of Given Angle Ref. Angle

45. Solution 2 1

46. Find the values of the six trig functions of the given angle in standard position.  Quadrantal Angles

49. Find the values of the six trig functions of the given angle in standard position. Quadrantal Angles

50. Find the values of the six trig functions of the given angle in standard position. Quadrantal Angles

51. Find the values of the six trig functions of the given angle in standard position. Quadrantal Angles

52. Important Ideas Trig functions of quadrantal angles have exact values. Trig functions of all other angles have approximate values. Trig functions of special angles have exact values.

53. Example Use a calculator to approximate cos to 4 decimal places. You did not forget to check “mode”, did you???

54. Example Use a calculator to approximate csc(11.55  ) to 4 decimal places. Check “mode” every time.

55. Lesson Close You will frequently use the trig knowledge you learned in Precalculus in Calculus

56. Assignment Worksheet 0-2