6-1 Angle Measures

1. 6-1 Angle Measures The Beginning of Trigonometry

2. The Radian The Radian is a new way to measure an angle. We are used to using degrees to indicate the size of an angle. Now we will use this new measuring tool. Its like being in Canada - the kilometer vs. the mile. The radian vs. the degree - either way, it indicates the specific size of an angle.

3. The Radian Definition: ______ ______________ ______________ ______________ ______________ ______________ ______________ ______________

4. Some Variables commonly used Arc Length = ________________ Central Angle =_______________ Radius =____________________ Now we’re going to see how these relate

5. I’m going to draw a picture with a central angle = 3 radians, then find the relationship between arc length, central angle, and radius. r r r r

6. What is the radian measure of a whole circle? For a whole circle, s = __________ = So, What does this mean? We now have a conversion factor (dimensional analysis)

7. Conversion Radians to Degrees: Degrees to Radians:

8. Examples: 1. Convert 55 to radians.

9. 2. Convert to degrees.

10. Area of a Sector Note: central angle should be in radians!!

11. Find the area of the sector of a circle with radius 4 if the central angle = 120.

12. 4. Find the area of the sector of a circle with radius 2 if the central angle =

13. 5. Find the area of the sector of a circle with radius 2 if the subtended arc = 6.

14. 6-2 Trigonometry

15. The Unit Circle Points Signs

16. θ

17. These are the reference triangles

18. Trigonometric Functions r y θ x

19. Trigonometric Functions r y θ x

20. Stuff I should know

21. How do I put this all together? 1. Determine what quadrant the terminal ray lies in and draw it. • Draw a line to the closest x axis (if you don’t, you will get the wrong answer) • ________________________________ ___________________________________ ___________________________________

22. Examples • Find What Quadrant?

23. Examples • Find What Quadrant?

24. Examples • Find Q IV What Quadrant?

25. What Happens if… If you land on an axis, there is no triangle. So instead of using sides of the triangle, use x, y and r. Unless otherwise specified, r = 1. • Find sin π, cos π and tan π. (-1,0)

26. 6-2 Trigonometry Day 2

27. Lets warm up…Solve the following

28. Yesterday we concentrated on sine, cosine and tangent functions. Today we will focus on the other 3 plus 3 trigonometric identities that you need to know!!!!

29. Trigonometric Functions r y θ x

33. Using Trig for other ‘s Find

34. Examples 1. When theta is placed in standard position , its terminal side passes thru the point (-2,2). Find all six trig Functions.

35. 2. When θ is put in standard position its terminal side passes through (-5, -12). Find all six trig functions of θ.

36. 6-3 Trigonometric Graphing Day 1 What do the graphs look like?

37. Sin x Lets plot some points x y 1 0 -1

38. Cos x x y 1 0 -1

39. Tan x x y 1 0 -1

40. Shifts Sin x 2 1 -1

41. Basic Truths The period (that is, cycle length) of both y = sin x and y = cos x is 2π. The amplitude of both functions is 1. For a standard, we start both of these functions at x = 0 and finish at x = 2π.

42. So what trends do we observe? Add outside? ________________ Add inside? ________________ Multiply outside? ________________ Multiply inside? ________________

43. Equation

44. Example y x 0

45. Now lets Graph it! 4 x y 1 -1 -2

46. Example y x 0

47. Now lets Graph it! x y 1 -1

48. 6-3 Trigonometric GraphingDay 2 Graphs

49. The other 4 Lets Remember Cot is the opp of? Csc is the opp of? Sec is the opp of?

50. How do trends hold here? They will hold pretty much the same way – seeing shifts left, right, up, down and seeing changes in periodicity.