Chapter 5 Trigonometric Functions of Angles

Transcription

1 Chapter 5 Trigonometric Functions of Angles

2 Section 3 Points on Circles Using Sine and Cosine

3 Signs

4 Signs I

5 Signs (+, +) I

6 Signs II (+, +) I

7 Signs II (, +) (+, +) I

8 Signs II (, +) (+, +) I III

9 Signs II (, +) (+, +) I III (, )

10 Signs II (, +) (+, +) I III (, ) IV

11 Signs II (, +) (+, +) I III (, ) (+, ) IV

12 Circles and Trig What was the equation of the unit circle?

13 Circles and Trig What was the equation of the unit circle? x + y = 1

14 Circles and Trig What was the equation of the unit circle? x + y = 1 What if we let x = cos(θ) and y = sin(θ).

15 Circles and Trig What was the equation of the unit circle? x + y = 1 What if we let x = cos(θ) and y = sin(θ). Does anyone remember what this is called when we make this substitution?

16 Circles and Trig What was the equation of the unit circle? x + y = 1 What if we let x = cos(θ) and y = sin(θ). Does anyone remember what this is called when we make this substitution? The Pythagorean Identity cos (θ) + sin (θ) = 1

17 Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ).

18 Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ). cos (θ) + sin (θ) = 1

19 Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ). cos (θ) + sin (θ) = 1 ( ) 1 cos (θ) + = 1 3

20 Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ). cos (θ) + sin (θ) = 1 ( ) 1 cos (θ) + = 1 3 cos (θ) = 1

21 Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ). cos (θ) + sin (θ) = 1 ( ) 1 cos (θ) + = 1 3 cos (θ) = 1 cos (θ) = 8 9

22 Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ). cos (θ) + sin (θ) = 1 ( ) 1 cos (θ) + = 1 3 cos (θ) = 1 cos (θ) = 8 9 cos(θ) = 3

23 Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5.

24 Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5. cos (θ) + sin (θ) = 1

25 Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5. cos (θ) + sin (θ) = 1 ( ) + sin (θ) = 1 5

26 Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5. cos (θ) + sin (θ) = 1 ( ) + sin (θ) = sin (θ) = 1

27 Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5. cos (θ) + sin (θ) = 1 ( ) + sin (θ) = sin (θ) = 1 sin (θ) = 1 5

28 Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5. cos (θ) + sin (θ) = 1 ( ) + sin (θ) = sin (θ) = 1 sin (θ) = sin(θ) = 5

29 Visually Speaking... (x, y) 1 θ

30 Visually Speaking... (x, y) θ 1 cos(θ)

31 Visually Speaking... (x, y) 1 sin(θ) θ cos(θ)

32 The Relationships ( (, 3, 1 ( 1, 0) ( 1 ), 3 ) ) 5π 6 π π 3 3π 4 π (0, 1) π 3 π 4 π ( ) 1, 3 π 6 ( ), ( 3, 1 ) (1, 0) ( 3 (, 1, ) ( 1, ) 3 ) 7π 6 5π 4 4π 3 3π 7π 4 5π 3 (0, 1) 11π 6 ( 3 (,, ) 1 ( 1, ) 3 )

33 Using the Unit Circle Find sin ( ) 3π 4.

34 Using the Unit Circle Find sin ( 3π 4 ).

35 Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ).

36 Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ). 1

37 Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ). 1 What angle has a sine of 1?

38 Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ). 1 What angle has a sine of 1? 7π 6, 11π 6

39 Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ). 1 What angle has a sine of 1? 7π 6, 11π 6 What angle has a cosine of 3?

40 Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ). 1 What angle has a sine of 1? 7π 6, 11π 6 What angle has a cosine of 3? π 6, 11π 6

41 Using Reference Angles We don t need to memorize the whole circle - just the first quadrant.

42 Using Reference Angles We don t need to memorize the whole circle - just the first quadrant. Notice that all of the values around this circle are ± 1, ± and ± 3. The ± decision is based on what quadrant the reference angle lies in.

43 Using Reference Angles Find sin ( ) 7π 6.

44 Using Reference Angles Find sin ( 7π 6 ). 7π 6

45 Using Reference Angles Find sin ( 7π 6 ). 7π 6

46 Using Reference Angles Find sin ( 7π 6 ). 7π 6 π 6

47 Using Reference Angles Find sin ( 7π 6 ). ( 3, 1 ) 7π 6 π 6

48 Using Reference Angles Find sin ( 7π 6 ). ( 3, 1 ) 7π 6 π 6 (, )

49 Using Reference Angles Find sin ( 7π 6 ). ( 3, 1 ) 7π 6 π 6 (, ) sin ( ) 7π 6 = 1

50 Using Reference Angles Find the angle(s) θ such that cos(θ) = 1.

51 Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y?

52 Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y? In what quadrants does cos(θ) have a positive value?

53 Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y? In what quadrants does cos(θ) have a positive value? In quadrant I, what angle are we looking for?

54 Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y? In what quadrants does cos(θ) have a positive value? In quadrant I, what angle are we looking for? π 3.

55 Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y? In what quadrants does cos(θ) have a positive value? In quadrant I, what angle are we looking for? π 3. What is the corresponding angle in quadrant IV?

56 Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y? In what quadrants does cos(θ) have a positive value? In quadrant I, what angle are we looking for? π 3. What is the corresponding angle in quadrant IV? π 3, or 5π 3.

57 Different Sized Circles What if we have a different radius?

58 Different Sized Circles What if we have a different radius? Different Radii The coordinate of a point on a circle of radius r corresponding to an angle θ is given by (rcos(θ), rsin(θ)).

59 Different Sized Circles What if we have a different radius? Different Radii The coordinate of a point on a circle of radius r corresponding to an angle θ is given by (rcos(θ), rsin(θ)). This is the exact same thing, except we multiply the value in question is multiplied by the radius to find the value.