1 / 102

Pre Calc Chapters 5 and 6

Pre Calc Chapters 5 and 6. Trigonometric Functions of Real Numbers. 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. The Unit Circle.

ginny
Download Presentation

Pre Calc Chapters 5 and 6

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

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

  9. 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!

  10. Reference Points

  11. Reference Points

  12. Reference Points

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

  14. 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!"

  15. Trig Functions

  16. Trig Functions Soh Cah Toa

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

  18. 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:

  19. Trig Functions

  20. Trig Functions

  21. Trig Functions: Using the Unit Circle

  22. Trig Functions: Using the Unit Circle

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

  24. 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

  25. Signs of Trig Functions

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

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

  28. 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.

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

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

  31. Graphing Trig Functions

  32. 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

  33. 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

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

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

  36. Sine Functions

  37. Cosine Functions

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

  39. p.439 #1-25 Odd

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

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

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

  43. Tangent

  44. Cot(x)

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

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

  47. Csc(x)

More Related