Pre Calc Chapters 5 and 6

1. Pre Calc Chapters 5 and 6 Trigonometric Functions of Real Numbers

2. The Unit Circle—5.1 Q: What do you get if you cross a mountain climber with a mosquito? A: Nothing, you can’t cross a scalar with a vector

3. The Unit Circle • The Unit Circle of radius 1 centered at the origin in the xy-plane. It’s equation is:

4. The Unit Circle Standard Position:

5. The Unit Circle • Is the point is on the unit circle??

6. The Unit Circle Measured in Radians: Why does 2π=360⁰ Think Circumference!

7. The Unit Circle Moving around the unit circle

9. The Unit Circle Terminal Point: The point P(x , y) obtained by traveling from the point (1 , 0)

10. Reference Points • The reference number is the shortest distance along the unit circle between the terminal point and the x-axis • Makes finding the corresponding ordered pairs easier to find!

11. Reference Points

12. Reference Points

13. Reference Points

14. p.416 #1-3, 5-7, 11-18, 23-25

15. Trigonometric Functions of Real Numbers—5.2 Three statisticians went duck hunting. A duck was approaching and the first statistician shot, and missed the duck by being a foot too high. The second shot and was a foot too low. The third cried, "We hit it!"

16. Trig Functions

17. Trig Functions Soh Cah Toa

18. Trig Functions • Let • Find sin(t) • Find cos(t) • Find tan(t)

19. Trig Functions Let t be any real number and let P(x,y) be a point on the unit circle: We can then define our trig functions as follows:

20. Trig Functions

21. Trig Functions

22. Trig Functions: Using the Unit Circle

23. Trig Functions: Using the Unit Circle

24. Trig Functions: Using the Unit Circle Using the point

25. Domain of Trig Functions • Sine, cosine • All Real Numbers • Tangent, Secant • All Real Numbers except for any integer n • Cotangent, Cosecant • All real numbers other than for any integer n

26. Signs of Trig Functions

27. Signs of Trig Functions All Students Take Calculus Sine All Tells us which values are positive Tangent Cosine

28. Even and Odd Functions • Even: • Sine, Cosecant, Tangent, and Cotangent • Odd: • Cosine and secant

29. So…What does that mean? • Tells us the sign of each function, based on the quadrant it is in! • Ex: consider • Sin, cos, tan, etc.

30. p.426 # 3-22, 27-29

31. Trig Functions—5.3 • There are 10 types of people in this world… …those who understand binary and those who don’t

32. Graphing Trig Functions

33. Periodic Functions • A function f is periodic iff there is a positive number p such that for every t • The smallest of these numbers p is called the period of the function

34. Periodic Functions • Think terminal points • …every 2pi units around the circle, you are at the same point so the function evaluated at those points will be the same

35. Sine and Cosine Curves • These two functions are often referred to as sinusoidal curves

36. Sine and Cosine Curves • General form of sine and cosine: • Amplitude: • Period: • phase shift: b • Vertical shift: c

37. Sine Functions

38. Cosine Functions

39. Can sin(x) be made to look like cos(x)?

40. p.439 #1-25 Odd

41. Trig Functions and Asymptotes—5.4 Q: How do we know the fractions are all European? A: Because they are all over C’s!

42. Tangent and Cotangent Graphs • Tangent and Cotangent both have a period of • In other words:

43. Where Do Asymptotes Occur? • At what values can Tan not be evaluated? • So we can say…

44. Tangent

45. Cot(x)

46. Modifying Graphs of Tan and Cot • Period • Amplitude • Phase Shift b • Vertical Shift c

48. Cosecant and Secant • csc and sec graphs have a period of • In other words:

50. Csc(x)