Angles and their measure

1. Angles and their measure Terminal side y Terminal side Vertex  x Initial Side Initial side Positive Angle in Standard Position Positive angles are generated by counterclockwise rotation. Negative angles are generated by clockwise rotation.

2. y Radian measure s = r r x r One radian is the measure of a central angle that intercepts an arc s equal in length to the radius r of thecircle.

3. Radian measure, continued. y s = 2r • It follows that a general angle has radian measure where s is the arc length intercepted by the angle. r x r Because the circumference of a circle is s = 2r units, it follows that the angle corresponding to a complete counterclockwise revolution is 2 radians.

4. Quadrants and angles Quadrant II Quadrant I Quadrant IV Quadrant III

5. Degree measure • Since a full counterclockwise revolution corresponds to 360º and also to 2 radians, we have • To convert degrees to radians, multiply degrees by • To convert radians to degrees, multiply radians by • Using we can also convert. For example, to find the degree measure for simply divide by 3 to get

6. Arc length • For a general angle, the radian measure is where s is the arc length intercepted by the angle and r is the radius of the circle. • If we multiply through by r, we obtain This is the formula for arc length, but it only applies when is given in radians. • Problem. What length of arc is cut off by an angle of 120 degrees on a circle of radius 12 cm? Solution. First convert to radians, then multiply by 12 to get an arc length of

7. Distance traveled by a rolling wheel of radius r = 1 • Suppose the wheel rolls without slipping as shown.

8. Effect of tire wear on mileage • The odometer in your car measures the mileage travelled. • The odometer uses the angle that the axle turns to compute the mileage s in the formula • Question. With worn tires, are the actual miles travelled more or less than the odometer miles?

9. Area of a sector of a circle Sector is shaded region • For a circle of radius r, the area A of a sector of the circle with central angle is given by where is measured in radians. • What area does A become when r

10. The unit circle The equation of the unit circle is y y x x On the unit circle, arc length = radian measure since r =1. Also, we write t for arc length instead of s.

11. Definitions of Trigonometric Functions   

12. Coordinates on unit circle for special angles         Divide unit circle into eight equal arcs as shown. Since (a,b) is on the line y = x, b equals a. Since (a, a) is on the unit circle, That is, which implies that Since the point is in the first quadrant, The coordinates of the other points are determined by symmetry.

13. Coordinates on unit circle for special angles, continued        

14. Coordinates on unit circle for special angles, continued             Divide unit circle into twelve equal arcs as shown. dist((a,b),(1,0)) = dist((a,b),(b,a)) implies b = 1/2. Since (a,b) is on the unit circle, Therefore, and Since (a,b) is in the first quadrant,

15. Coordinates on unit circle for special angles, continued            

16. Domain, range, and period of sine and cosine • Domain of sine and cosine: All real numbers • Range of sine and cosine: [–1, 1] • A function f is periodic if there exists a positive real number c such that for all t in the domain of f. The smallest number c for which f is periodic is called the period of f. • Since it follows that sine and cosine are periodic with period 2.

17. More properties of trigonometric functions • The cosine and secant functions are even: • The sine, cosecant, tangent, and cotangent functions are odd: • Problem. Evaluate Solution. Since • Problem. Solution.

18. Definition of Sine and Cosine in Right Triangles Suppose P = (x, y) is the point on the unit circle specified by the angle We define the functions, cosine of , or , and the sine of , or by either pair of formulas: Hypotenuse (-1,0) Side Opposite Side Adjacent

19. Definition of the tangent function Suppose P = (x, y) is the point on the unit circle specified by the angle We define the tangent of , or tan , by tan = y/x, or by: Side Opposite Side Adjacent

20. Evaluating trigonometric functions Evaluate sine, cosine, and tangent for the triangle shown. • First, hyp 5 12

21. Another way to derive the trig functions of special angles φ h • Consider the following right triangle. • Evaluate: 1 θ 90º 1

22. More on trig functions of special angles • Consider the following equilateral triangle. • Evaluate: 2 2 h α β 1 1

23. Sines, cosines, and tangents of special angles

24. Fundamental Trigonometric Identities • Reciprocal identities • Quotient identities • Pythagorean identities

25. Evaluating trigonometric functions using trig identities Evaluate cosine for the triangle shown. • Using trig identities: hyp 5 12

26. Solving Right Triangles • One of the angles of a right triangle is 90°. Thus, one part is already known and the only additional information necessary is either two sides or an acute angle and a side. With this additional information, it is possible to solve for the remaining parts. • Making a carefully labeled sketch of the triangle is important in solving these problems. • The top of a 200-foot tower is to be anchored by cables that make an angle of 30° with the ground. How long must the cables be? How far from the base of the tower should the anchors be placed? sin 30° = 200/h => h = 200/sin 30° = 400 ft tan 30° = 200/x => x = 200/tan 30° = 346.4 ft h 200 ft 30° x

27. Height of church steeple • At a point 65 ft. from the base of a church, the angles of elevation to the bottom of the steeple and the top of the steeple are 35º and 43º, resp. Find the height s of the steeple. • s = h2 – h1 = 65tan 43º– 65tan 35º s h2 h1 43º 35º 65

28. Definitions of trigonometric functions of any angle y Let be an angle in standard position with (x, y) a point onthe terminal side of and x

29. Interpretation of "any angle" definition If we replace by similar right triangles may be used and • The "any angle" definition is the same as the "unit circle" definition but on a circle of radius r   Since both x and are negative Thus, cos has the same value by either definition.

30. Reference Angles • Let θbe an angle in standard position. Its reference angle is the acute angle θ' formed by the terminal side of θand the horizontal axis. • Example. Find the reference angle of

31. Evaluating trigonometric functions of any angle θ • Determinethe function value for the associated reference angle θ'. • Depending on the quadrant in which θlies, affix the appropriate sign to the function value. • Example. Find the value of cos 135º.

32. The Unit Circle For any angle : (x,y) = (cos , sin ) y x

33. Evaluating trig functions • Let θbe an angle in Quadrant III such that sin θ = Find the value of cos θusing trig identities. • Using sin2θ + cos2θ = 1, we have It follows that Since θis in Quadrant III, • It is also possible to solve for cos θ using a reference angle. See the next slide.

34. Evaluating trig functions using a reference angle Find cos θfor the angle θshown where sin θ = By Pythagoras on the triangle, So, |x| = 12. Therefore, cos θ' =12/13 and cos θ = –12/13since θis in QIII.

35. Sines and cosines of special angles Using these values, we can plot the basic sine and cosine curves.

36. Basic sine curve: Any portion of the graph representing one period is called one cycle. y     x 

37. Basic cosine curve: Any portion of the graph representing one period is called one cycle. y     x 

38. Sketching basic sine and cosine graphs using five key points The five key points in one period of a graph: intercepts, maximum points, and minimum points. y  x     y   x   

39. Transformations of sine and cosine curves • Consider the functions defined by • We will consider the effects of the constants a, b, c, and d on the shape of the basic sine and cosine curves. • First, the constant a determines a vertical stretch (a > 1) or a vertical shrink (0 < a < 1) of the basic sine and cosine curves. • |a| is the amplitude, which is half the distance between the maximum and minimum values of the functions *. *

40. Example for vertical stretch, vertical shrink Consider the functions defined by y x

41. Example for horizontal stretch • Consider the function defined by • The transformed graph will exhibit a horizontal stretch and the new period will be 4. • In general, will have a period of y y x

42. Horizontal translation of the sine and cosine curves • The constant c in the general equations creates a horizontal translation of the sine and cosine curves. • The graphs of have the following characteristics. (Assume b > 0.) The left and right endpoints of a one-cycle interval can be determined by solving the equations • The graphs of and are shifted by an amount c/b in their respective general equation. The number c/b is called the phase shift.

43. Example for horizontal translation • Consider the function defined by y = sin(x–/2). • The amplitude is 1, the period is 2, and the phase shift is 1/2. • x–/2 = 0 => x = 0.5, x–/2 = 2 => x = 2.5 so the interval [0.5, 2.5] corresponds to one cycle of the graph. y y = sin(x–/2) y = sin x x

44. Sine with a phase shift of –π/2 • plot([sin(t),sin(t+Pi/2)], t = –Pi..Pi, color = black); • The shifted sine curve is the cosine. Likewise, we can shift the cosine curve right by π/2 to get the sine.

45. Vertical translation of the sine and cosine curves • The constant d in the general equations creates a vertical translation of the sine and cosine curves. The line y = d is called the midline of the translated curve. Note that amplitude = distance from midline to max value = distance from midline to min value. • Consider the function defined by y =0.5+cos x. amplitude = 1 midline is y = 0.5 x

46. Graph of a sine function. (a) What is the period? (b) What is the midline? (c) What is the amplitude? (d) Give a possible formula for this sine function.

47. Match the graph and the equation: (ii) (iii) (i) (vi) (v) (iv) (a) y = 2(sin 2t) (b) y = 2(cos 2t) (c) y = 2(cos t) + 2 (d) y = 2sin (t + 2) (e) y = 2(sin t) + 2 (f) y = 2(sin 2t) +2

48. London Eye Ferris wheel • The London Eye Ferris wheel has a diameter of 450 feet. It completes one revolution every 30 minutes, and boarding is from ground level. As a function of angle θ, the height y in feet of a seat on this Ferris wheel is where θis in radians. y θ   ground level

49. London Eye Ferris wheel, continued • As a function of time t in minutes, the height y in feet of a seat on this Ferris wheel is ground level

50. Graphs of tangent and cotangent