1. 1 Trigonometric Models Chapter 9 Trigonometric Functions, Models, and Regressions (Lecture 21, Wed. 3/28/07) Derivatives of Trigonometric Functions and Applications (Lecture 22, Mon. 4/2/07) Integrals of Trigonometric Functions and Applications (Lecture 23, Wed. 4/4/07) Lecture 21

2. 2 Trigonometric Functions, Models, and Regressions Examples: Sunspot Activity, Piston in a Cylinder Tides, AC Current and AC Voltage Average Daily Air Temperature Seasonal Fluctuations in Business Demand Hair Cut/Growth, Fluffy Paces on the Ledge, Sound Waves, …etc.

3. 3 Calculating sin(  ), cos(  ), and tan(  ) Triangle method: SOH, CAH, TOA  hypotenuse (length = h) adjacent side (length = x) opposite. side (length = y) “SOH” sin(  ) = (opposite side)/hypotenuse“SOH” sin(  ) = (opposite side)/hypotenuse “CAH” cos(  ) = (adjacent side)/hypotenuse“CAH” cos(  ) = (adjacent side)/hypotenuse “TOA” tan(  ) = (opposite side)/(adjacent side)“TOA” tan(  ) = (opposite side)/(adjacent side)

4. 4 The Sine Function The sine of a real number t is the y–coordinate (height) of the point P in the following diagram, where |t| is the length of the arc. x y P sin t 1 unit 1 1 –1 |t||t|

5. 5 The Sine Function -1.5 -0.5 0 0.5 1 1.5 -1.5-0.50.511.5 Highlight those sections of the circle where sin(t) >0 sin(t) > 0

6. 6 The Sine Function

7. 7 The General Sine Function A is the amplitude (peak height above baseline) C is the vertical offset (height of baseline) P is the period (wavelength) is the angular frequency is the phase shift

8. 8 Sine Function 1.5 is the amplitude 1.8 is the vertical offset is the period 0.5 is the angular frequency 2.6 is the phase shift Example:

9. 9 Sine Function Example: Basepoint

10. 10 The Cosine Function The cosine of a real number t is the x– coordinate (length) of the point P in the following diagram, where |t| is the length of the arc. x y P cos t 1 unit 1 1 –1 |t||t|

11. 11 The Cosine Function -1.5 -0.5 0 0.5 1 1.5 -1.5-0.50.511.5 cos(t) > 0 Highlight those sections where cos(t) > 0

12. 12 The Cosine Function

13. 13 The General Cosine Function A is the amplitude (peak height above baseline) C is the vertical offset (height of baseline) P is the period (wavelength) is the angular frequency is the phase shift

14. 14 Fundamental Trigonometric Identities (Relationships Between Cosine and Sine) -1.5 -0.5 0 0.5 1 1.5 -1.5-0.50.511.5 Alternative:

15. 15 Other Trigonometric Functions Tangent: Cotangent: Secant: Cosecant:

16. 16 Other Trignometric Functions The Tangent and Cotangent Functions y= tan(x) tan(x) = sin(x)/cos(x) y = cot(x) cot(x) = cos(x)/sin(x) = 1/tan(x)

17. 17 Trigonometric Regression Use data that suggests a sine (or cosine) curve and perform a regression to find the best-fit generalized sine (or cosine) curve.

18. 18 Example: Cash Flows into Stock Funds The annual cash flow into stock funds (measured as a percentage of total assets) has fluctuated in cycles of approximately 40 years since 1995, when it was at a high point. The highs were roughly +15% of total assets, whereas the lows were roughly  10% of total assets. a.Model this cash flow with a cosine function of the time t in years, with t = 0 representing 1995. b.Convert the answer in part (a) to a sine function model.

19. 19 Example: Cash Flows into Stock Funds Solution:

20. 20 Another Example of Periodic Systems Sunspot Counts

21. 21 Section 9.1 Problem #39 (Waner pp. 552  553) Sunspot Activity  The activity of the Sun (sunspots, solar flares, and coronal mass ejection) fluctuates in cycles which can be modeled by a.What is the period of sunspot activity according to this model? b.What is the maximum number of sunspots observed? What is the minimum number? c.When to the nearest year, is sunspot activity next expected to reach a high point? Where t is the number of years since January 1, 1997, and N(t) is the number of sunspots observed at time t.

22. 22 Section 9.1 Problem #39 (Waner pp. 552  553) Solution: