Chapter 4 – Trigonometric Functions

1. Chapter 4 – Trigonometric Functions

2. 4.1 – Angles and Their MeasuresDegrees and Radians • Degree – denoted ̊ , is a unit of angular measure equal to 1/180th of a straight angle. • In the DMS (____________________) system of angular measure, each degree is subdivided into 60 minutes (denoted by ’) and each minute is subdivided into 60 seconds (denoted by ’’ ).

3. Working with DMS measure • (a) Convert 37.425 ̊ to DMS • (b) Convert 42 ̊24’36’’ to degrees.

4. Radian • A central angle of a circle has measure 1 radian if it intercepts an arc with the same length as the radius.

5. Working with Radian Measure • How many radians are in 90 degrees? • How many degrees are in /3 radians?

6. Cont… • Find the length of an arc intercepted by a central angle of ½ radian in a circle of radius 5 inches.

7. Degree-Radian Conversion • To convert radians to degrees, multiply by • To convert degrees to radians, multiply by

8. Practice: • Work with partners • Pg 356 # 1-24e

9. Do Now: • Convert from radians to degrees: • π/6 • π/10 • 5π/9

10. Circular Arc Length • Arc Length Formula (Radian Measure) • If θ is a central angle in a circle of radius r, and if θ is measure in radians, then the length s of the intercepted arc is given by • S=rθ • Arc Length Formula (Degree Measure) • If θ is a central angle in a circle of radius r, and if θ is measured in degrees, then the length s of the intercepted arc is given by • S=πrθ/180

11. Find the perimeter of a 60° slice of a large pizza (with 8 in radius)

12. 4.2 - Trigonometric Functions of Acute Angles HW: Pg 368 #2-32e

13. Right Triangle Trigonometry - DEF: Trigonometric Functions • Let be an acute angle in the right ABC. Then • Sine θ= sinθ = • Cosineθ = cosθ = • Tangentθ = tanθ = • Cosecantθ = cscθ = hyp/opp • Secantθ = secθ = hyp/adj • Cotangentθ = cotθ = adj/opp

14. Find the values of all six trigonometric functions for an angle of 45°

15. Find the values of all six trigonometric functions for an angle of 30º

16. Let θ be an acute angle such that sin θ = 5/6. Evaluate the other five trigonometric functions of θ.

17. Solving a Right Triangle • A right triangle with a hypotenuse of 8 includes a 37 ̊ angle. Find the measures of the other two angles and the lengths of the other two sides.

18. From a point 340 feet away from the base of the Tower of Weehawken, the angle of elevation to the top of the building is 65 ̊. • Find the height h of the building.

19. 4.3 - Trigonometry Extended - The Circular Functions HW: Pg. 381 #2-6e, 26-36e

20. Trigonometric Functions of Any Angle • Positive angles are generated by counterclockwise rotations • Negative angles are generated by clockwise rotations.

21. Standard Position • Vertex - • Initial side -

22. Coterminal Angles • Two angles can have the same initial side and the same terminal side, yet have different measures - called coterminal angles.

23. Find Coterminal Angles • 40 ̊ • -160 ̊ • 2/3 radians

24. Trigonometry in Quadrant I: • Let P(x,y) be any point in the first quadrant, let r be the distance from P to the origin. • Use the acute angle definition to show the sinθ = y/r 2. Express cos θ= in terms of x and r. 3. Express tan θ= in terms of x and y. 4. Express the remaining three basic trigonometric functions in terms of x, y, and r. y P(x,y) r θ x

25. Evaluating Trig Functions Determined by a Point in QI • Letθ be the acute angle in standard position whose terminal side contains the point (4,3). Find the six trigonometric functions of θ.

26. Evaluating Trig Functions Determined by a Point NOT in QI • Letθ be an angle in standard position whose terminal side contains the point (-5,2). Find the six trigonometric functions of θ.

27. Trigonometric Functions of any Angle • Let θ be any angle in standard position and let P(x,y) be any point on the terminal side of the angle (except the origin). Let r denote the distance from P(x,y) to the origin, i.e., Let r = √(x2+y2). Then • Sin θ= • cosθ= • tanθ= • cscθ= • secθ= • cotθ= P(x,y) y r θ x

28. Evaluating the Trig Functions of 315 ̊ • Find the six trigonometric functions of 315 ̊

29. Evaluating More Trig Functions • Sin(-210 ̊) • Tan(5/3) • Sec(-3/4)

30. … • Sin(-270 ̊) • Tan(3) • Sec(11/2)

31. When are sin, cos, and tan positive? • Quadrant 1- • Quadrant II – • Quadrant III – • Quadrant IV –

32. Using One Ratio to Find the Others Find cos θ and tan θ if: • Sinθ = 3/7 and tanθ <0 • secθ=3 and sinθ>0 • cotθ is undefined and secθ is negative

33. …Try on your own • Find sinθ and tanθ if cosθ=2/3 and cotθ>0 • Find tanθ and secθ if sinθ=-2/5 and cosθ>0 • Find secθ and cscθ if tan θ=-4/3 and sinθ>0

34. Trigonometric Functions of Real Numbers • DEF: Unit Circle • The unit circle is a circle of radius 1 centered at the origin.

35. Wrapping Function • Connects points on a number line with points on the circle

36. Trigonometric Functions of Real Numbers • Let t be any real number, and let P(x,y) be the point corresponding to t when the number line is wrapped onto the unit circle as described above. Then • Sin t = • Cos t = • Tan t = • Csc t = • Sec t = • Cot t = • Therefore, the number t on the number line always wraps onto the point (cos t, sin t) on the unit circle P(cos t, sin t)

37. Exploring the Unit Circle • For any t, the value of cos t lies between -1 and 1 inclusive. • For any t, the value of sin t lies between -1 and 1 inclusive. • The values of cos t and cos (-t) are always opposites of each other. (Recall that this is the check for an even function.) • The values of sin t and sin(-t) are always opposites of each other. (Recall that this is the check for an odd function.) • The values of sin t and sin (t +2) are always equal to each other. In fact, that is true of all six trig functions on their domains, and for the same reason. • The values of sin t and sin (t+) are always opposites of each other. The same is true of cost and cos (t+) • The values of tan t and tan (t+) are always equal to each other (unless they are both undefined). • The sum (cost)2 + (sint)2 always equal 1.

38. Periodic Functions • Def: Periodic Functions • A function y=f(t) is periodic if there is a positive number c such that f(t+c) = f(t) for all values of t in the domain of f. The smallest such number c is called the period of the function.

39. Using Periodicity • Sin (57801/2)

40. Cont.. • Cos(288.45) – cos(280.45) • Tan(/4 – 99,999)

41. 16-Point Unit Circle

42. The 16- Point Unit Circle

43. 4.4 – Graphs of Sine and Cosine: Sinusoids HW: Pg. 393-394 #1-16e, 50-56e

44. Graph of: F(x)=sinx F(x)=cosx

45. Sinusoids and Transformations • A function is a sinusoid if it can be written in the form • F(x) = a sin(bx+c)+d Where a, b, c, and d are constants and a,b≠0.

46. Transformations • Horizontal Stretches and Shrinks affect the period and the frequency • Vertical Stretches and Shrink affect the amplitude • Horizontal translations bring about phase shifts

47. Amplitude of a Sinusoid • The amplitude of the sinusoid f(x)=asin(bx+c)+d is |a| • F(x)=acos(bx+c)+d • Graphically, the amplitude is half the height of the wave

48. Vertical Stretch or Shrink and Amplitude • Find the amplitude of each function and describe how the graphs are related • Y1=cosx • Y2= 1/2cosx • Y3 = -3cosx

49. Period of a Sinusoid • The period of the sinusoid f(x)=asin(bx+c)+d is 2/|b|. • Graphically, the period is the length of one full cycle of the wave.

50. Find the period of each function and describe how the graphs are related: • Y1= sinx • Y2= -2sin(x/3) • Y3= 3sin(-2x)