1. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §8.1 Angles & TrigoNometry

2. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §7.6 → Double Integrals  Any QUESTIONS About HomeWork §7.6 → HW-9 7.6

3. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 3 Bruce Mayer, PE Chabot College Mathematics Angles: Basic Terms  Two distinct points determine a line called Line AB  Line segment AB → a portion of the line between A and B, including points A and B.  Ray AB → a portion of line AB that starts at A and continues through B, and on past B A B AB A B

4. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 4 Bruce Mayer, PE Chabot College Mathematics Angles: Basic Terms  Angle: formed by rotating a ray around its endpoint.  The ray in its initial position is called the initial side of the angle  The ray in its location after the rotation is the terminal side of the angle

5. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 5 Bruce Mayer, PE Chabot College Mathematics Identifying Angles  Unless it is ambiguous as to the meaning, angles may be named only by a single letter (English or Greek) displayed at vertex or in area of rotation between initial and terminal sides  Angles may also be named by three letters, one representing a point on the initial side, one representing the vertex and one representing a point on the terminal side (vertex letter in the middle, others first or last)

6. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 6 Bruce Mayer, PE Chabot College Mathematics Positive & Negative Angles  Positive angle: The rotation of the terminal side of an angle counterclockwise.  Negative angle: The rotation of the terminal side is clockwise. Positive AngleNegative Angle

7. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 7 Bruce Mayer, PE Chabot College Mathematics Angle: Measures & Classes  The most common unit for measuring angles is the degree (°) One Rotation or Cycle = 360°  Four Classes of Angle: Acute, Right, Obtuse, Straight

8. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 8 Bruce Mayer, PE Chabot College Mathematics Angle: RADIAN Measure  Define the “Radian” measure as the SubTended Circumferential distance on a circle divided by the radius.  Thus a subtended angle that produces an arc-length of 1 radius is 1 radian in measure  Radians in one Cycle:

9. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 9 Bruce Mayer, PE Chabot College Mathematics Degrees & Radians Compared MeasureDescriptionGraphic One Quarter Revolution One Half Revolution Three Quarter Revolution One Full Revolution

10. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 10 Bruce Mayer, PE Chabot College Mathematics Degrees ↔ Radians  The Measure of One Cycle  Then the Number “1”  Convert to other Measure: 53°, 2.2 rad

11. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 11 Bruce Mayer, PE Chabot College Mathematics Unit Circle  Imagine a circle on the CoOrdinate plane, with its center at the origin, and a radius of 1.  Choose a point on the circle somewhere in quadrant I.

12. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 12 Bruce Mayer, PE Chabot College Mathematics Unit Circle  Connect the origin to the point, and from that point drop a perpendicular to the x-axis.  This creates a right triangle with hypotenuse of 1.

13. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 13 Bruce Mayer, PE Chabot College Mathematics Unit Circle  The length of its legs are the x and y coordinates of the chosen point.  Applying the definitions of the trigonometric ratios to this triangle gives x y 1  is the angle of rotation

14. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 14 Bruce Mayer, PE Chabot College Mathematics Unit Circle  Thus The CoOrdinates of the chosen point are the CoSine (x) and Sine (y) of the angle  This provides a way to define functions sin(  ) and cos(  ) for all real numbers  The Four other trigonometric functions can be defined from the Unit Circle as well

15. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 15 Bruce Mayer, PE Chabot College Mathematics The 16-Point Unit Circle

16. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 16 Bruce Mayer, PE Chabot College Mathematics Unit Circ Tabulated

17. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Calc Sin & CoSin  Find the values:  Negative angles are represented by traversing the Unit Circle ClockWise, so the terminal side of an angle of −π/2 rads (−90°) falls on the negative y-axis and takes the point (1,0) to the point (0,−1).  The CoSine is given by the x-coordinate at this point, so

18. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Calc Sin & CoSin  SOLUTION:  The terminal side of the angle with measure 5π/4 rads (225°) falls on the line in the third quadrant which takes the point (1,0) to the point:  The Sine is the y-coordinate of this point, so

19. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 19 Bruce Mayer, PE Chabot College Mathematics Graph: Sine & CoSine

20. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 20 Bruce Mayer, PE Chabot College Mathematics Properties of Sine & CoSine  From the Periodic Nature of the Sinusoidal Graphs Observe

21. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 21 Bruce Mayer, PE Chabot College Mathematics MATLAB Code % Bruce Mayer, PE % MTH-16 22Feb14 % MTH15_Quick_Plot_BlueGreenBkGnd_130911.m % clear; clc; clf; % clf clears figure window % % The Domain Limits xmin = -4*pi; xmax = 4*pi; % The FUNCTION ************************************** x = linspace(xmin,xmax,1000); y = sin(x); y1 = cos(x); % *************************************************** % the Plotting Range = 1.05*FcnRange ymin = min(y); ymax = max(y); % the Range Limits R = ymax - ymin; ymid = (ymax + ymin)/2; ypmin = ymid - 1.025*R/2; ypmax = ymid + 1.025*R/2 % % The ZERO Lines zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ypmin*1.05 ypmax*1.05]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green subplot(2,1,1) plot(x,y, 'LineWidth', 4),grid, axis([xmin xmax ypmin ypmax]),... xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = sin(\theta)'),... title(['\fontsize{16}MTH16 sin(\theta)']) hold on plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) hold off subplot(2,1,2) plot(x,y1, 'LineWidth', 4),grid, axis([xmin xmax ypmin ypmax]),... xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = cos(\theta)'),... title(['\fontsize{16}MTH16 cos(\theta)',]) hold on plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) hold off

22. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 22 Bruce Mayer, PE Chabot College Mathematics Trig Fcn RelationShips  4 of the 6 Trig Functions can be expressed in Terms of the basis functions of sin and cos  With reference to the Unit Circle Find

23. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 23 Bruce Mayer, PE Chabot College Mathematics Pythagorean Identities  ReCall the Pythagorean Theorem  The Unit Circle Analogy

24. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 24 Bruce Mayer, PE Chabot College Mathematics Pythagorean Identities  Also  In Summary

25. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  Use Trig Relns  Find the value of cos(θ) given that csc(θ) = 3 the angle θ is contained in a right triangle  SOLUTION:  Recall from Unit Circle:  Next use the Pythagorean Identity

26. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  Use Trig Relns  Then in This case  So  But since θ is confined to right triangle θ must be less than 90° then the cos must be POSITIVE  Thus if csc(θ) = 3, then

27. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 27 Bruce Mayer, PE Chabot College Mathematics Example  Sinusoidal Periodicity  A math Model for the Diurnal hours of daylight t months after January 1 in Eugene, Oregon  Use this model to Find the amplitude, period, horizontal and vertical shifts of the function. –Interpret the values

28. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 28 Bruce Mayer, PE Chabot College Mathematics Example  Sinusoidal Periodicity  SOLUTION:  The amplitude is the distance from average to high (or average to low) values of the function. This is represented by the absolute value of the CoEfficient on the trigonometric function (sine in this case). amplitude

29. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example  Sinusoidal Periodicity  Thus by the sinusoidal amplitude over time, the daylight hours in Eugene varies 3.17 up & down from its average.  SOLUTION:  The period of a sine function is the value p when written in the form

30. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example  Sinusoidal Periodicity  Factor to produce a t-CoEfficient of this form in the given function-argument:  Then by sine-argument Correspondence  The function repeats itself every 11.64 months, which is probably a rough approximation of the 12-month yearly cycle of daylight

31. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 31 Bruce Mayer, PE Chabot College Mathematics Example  Sinusoidal Periodicity  SOLUTION:  The horizontal shift (also called the Phase-Shift) of the function is given by the value of d in the form  Again by sine-argument Correspondence

32. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 32 Bruce Mayer, PE Chabot College Mathematics Example  Sinusoidal Periodicity  The d = 2.8 months suggests that the average value is not achieved at t = 0 (December 31st), but rather the function is close to its minimum in early spring, about 2.8 months in to the Year.  SOLUTION:  The vertical shift (also called the mean value) of the function is given by the value of a in the form  Again by sine-argument Correspondence

33. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 33 Bruce Mayer, PE Chabot College Mathematics Example  Sinusoidal Periodicity  Then by function Correspondence  The function does not vary equally above and below zero (negative daylight hours makes no sense). Instead, the average value is 12.2 hours and the function varies up and down from that midline.

34. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 34 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §8.1 P8.1-68 → Home Heating Energy Use in Buffalo, NewYork 2010 Weather SummaryBuffalo, NYNew YorkUSA Weather Index2229100 Hail Index339100 Hurricane Index3493100 Tornado Index4033100 Annual Maximum Avg. Temperature56.0 °F57.0 °FN/A Annual Minimum Avg. Temperature40.0 °F39.0 °FN/A Annual Avg. Temperature47.7 °F N/A Annual Heating Degree Days (Tot Degrees < 65)6,7476,762N/A Annual Cooling Degree Days (Tot Degrees > 65)477484N/A Percent of Possible Sunshine4851N/A Mean Sky Cover (Sunrise to Sunset - Out of 10)77N/A Mean Number of Days Clear (Out of 365 Days)5465N/A Mean Number of Days Rain (Out of 365 Days)169150N/A Mean Number of Days Snow (Out of 365 Days)2621N/A Avg. Annual Precipitation (Total Inches)39.00"38.00"N/A Avg. Annual Snowfall (Total Inches)91.00"75.00"N/A

35. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 35 Bruce Mayer, PE Chabot College Mathematics All Done for Today More Trig Identities

36. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 36 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

37. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 37 Bruce Mayer, PE Chabot College Mathematics

38. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 38 Bruce Mayer, PE Chabot College Mathematics

39. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 39 Bruce Mayer, PE Chabot College Mathematics