Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §8.1 Angles &TrigoNometry Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 7.6 Review § • Any QUESTIONS About • §7.6 → Double Integrals • Any QUESTIONS About HomeWork • §7.6 → HW-9

3. A B A B A B Angles: Basic Terms • Two distinct points determine a line called LineAB • Line segment AB → a portion of the line between A and B, including points A and B. • Ray AB → aportion of line AB that starts at A and continues through B, and on past B

4. Angles: Basic Terms • Angle: formed by rotating a ray around its endpoint. • The ray in its initial position is called the initial sideof the angle • The ray in its location after the rotation is the terminal side of the angle

5. 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. 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 Angle Negative Angle

7. 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. Angle: RADIAN Measure • Define the “Radian” measure as the SubTended Circumferential distance on a circle divided by the radius. • Thus a subtendedangle that produces anarc-length of 1 radius is 1 radian in measure • Radians inone Cycle:

9. Degrees & Radians Compared

10. Degrees ↔ Radians • The Measure of One Cycle • Then the Number “1” • Convert to other Measure: 53°, 2.2 rad

11. 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. 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. Unit Circle  is the angle of rotation • The length of its legs are the xand y coordinates of the chosen point. • Applying the definitions of the trigonometric ratios to this triangle gives 1 y x

14. 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. The 16-Point Unit Circle

16. Unit Circ Tabulated

17. 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. 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. Graph: Sine & CoSine

20. Properties of Sine & CoSine • From the Periodic Nature of the Sinusoidal Graphs Observe

21. % 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 = [xminxmax]; 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([xminxmaxypminypmax]),... 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([xminxmaxypminypmax]),... 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 MATLAB Code

22. Trig FcnRelationShips • 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. Pythagorean Identities • ReCall the Pythagorean Theorem • The Unit Circle Analogy

24. Pythagorean Identities • Also • In Summary

25. 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. 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. 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. 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. 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. 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 11.64months, which is probably a rough approximation of the 12-month yearly cycle of daylight

31. 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. 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. 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. WhiteBoard Work • Problems From §8.1 • P8.1-68 → HomeHeating EnergyUse in Buffalo, NewYork

35. All Done for Today MoreTrigIdentities

36. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –