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S4 Credit Chapter 5

TRIGONOMETRIC GRAPHS

MAXIMUM 4 x 1 + 6 = 10 when 3x = 90x = 30

MINIMUM 4 x (-1) + 6 = 2 when 3x = 270x = 90

maximum value 10 when x = 30 or maximum turning point ( 30,10 )minimum value 2 when x = 90 or minimum turning point ( 90, 2 )

X

Y

0

c + a

c

The graph is of the form y = a sin b x° + c.

c - a

360 b

270 b

180 b

90 b

a

by 360 3

270 3

180 3

90 3

X

Y

0

10

6

The graph is y = 4 sin 3 x° + 6.

2

120906030

4

X

Y

8

-8

350-10 170

The graph is y = 8 sin( x +10)°.

80 260

X

Y

5

-5

37010 190

The graph is y = 5 sin(x -10)°.

100 280

X

Y

a

-a

360 - b-b 180 - b

The graph is of the form y = a sin( x + b)°.

90 - b 270 - b

X

Y

0

8

5

360

2

y = 4A B

y = 5 + 3 cos x°

The graphs with equations y = 5 + 3 cos x° and y = 4 are shown.Find the x co-ordinates of the points of intersection A and B.

5 + 3 cos x° = 4 3 cos x° = -1

cos x° = -1/3x = 109.5 or x = 250.5

AS

T C

360 - a = 289.5°

a = cos -1 1/3 = 70.528...°180 - a = 109.5°

180 + a = 250.5°

A, S, T, C where functions positive:

cos +

cos +

cos -

cos -

X

Y

360°0°

6

-6

The graph is of the form y = a sin b x. Find the values of a and b.1.

X

Y

360°0°

10

-10


X

Y

360°0°

3

-3


X

Y

360°0°

8

-8

The graph is of the form y = a cos b x. Find the values of a and b.4.

X

Y

360°0°

16

-16


X

Y

360°0°

12

-12


X

Y

120°0°

5

-5


X

Y

0°

9

-9

30°


X

Y

72°0°

11

-11


The graph is of the form y = a cos b x. Find the values of a and b.

X

Y

18

-18

0° 120°60°30° 90°

10.


X

Y

0°

4

-4

60°30°15° 45°

11.


X

Y

14

-14

0° 40°20°10° 30°

12.

X

Y

180°0°

2.5

-2.5


X

Y

360°0°

7

-7


X

Y

360°0°

5

-5


X

Y

720°0°

3.2

-3.2


X

Y

360°0°

6

-6


X

Y

360°0°

8

-8


X

Y

3

-3

355-5 175

The graph is of the form y = a sin(x+b)°. Find the values of a and b.19.

X

Y

4

-4

3688 188


X

Y

7

-7

350-10 170


The graph is of the form y = a cos(x+b)°. Find the values of a and b.

-3.6

X

Y

3.6

2900 20011020 380

22.


X

Y

0

2

-2

33515565 245

23.


-6.5

6.5

2820 19210212 X

Y24.

120° X

Y

0°

45

5

25

The graph is of the form y = a sin b x + c. Find the values of a , b and c.25.

X

Y

30°0°

50

25

70° 110° 150° 190°

The graph is of the form y = a sin b x + c. Find the values of a , b and c.26.

X

Y

0

25

6 12

5

15

The graph is of the form y = a sin b x° + c. Find the values of a , b and c.27.

The graph is of the form y = a cos b x° + c. Find the values of a , b and c.

X

Y

0

75

45

20 40

15

10 30

28.

The graph is of the form y = a cos b x + c. Find the values of a , b and c.Y

20

37.5

X15°0° 45°2.5

29.

The graph is of the form y = a cos b x + c. Find the values of a , b and c.

32

10

-12

X

Y

27°0° 18°9°

30.

For questions 31 to 36 find the co-ordinates of the maximum and minimum turning points of the graph.

min.

X

Y

0

max.

20

15

The graph has equation y = 15 + 10 sin18 x°.31.

32. The graph has equation y = 120 + 100 cos30 x°.

X

Y

120°

max.

min.

120

min.

X

Y

0

max.

24

10

The graph has equation y = 10 + 20 sin15 x°.33.

The graph has equation y = 15 sin( 9 x - 18)° + 20.

min.

X

Y

0

max.

40

34.

The graph has equation y = 100 + 60 cos( 5 x - 15)°.

min.

X

Y

0

max.

72

35.

The graph has equation y = 50 + 140 cos(10 x + 12)°.36.

X

Y

0

max.

min.

36

For questions 37 to 51 give your answers correct to one decimal place.

X

Y

360°0°

5

-5

y = 4

y = 5 sin xA B

The graphs with equations y = 5 sin x and y = 4 are shown.Find the x co-ordinates of the points of intersection A and B.

37.

X

Y

360°0°

6

-6

y = 6 sin x

y = 5

A B

The graphs with equations y = 6 sin x and y = 5 are shown.Find the x co-ordinates of the points of intersection A and B.

38.

X

Y

360°0°

3

-3

y = 1

y = 3 cos x

A B

The graphs with equations y = 3 cos x and y = 1 are shown.Find the x co-ordinates of the points of intersection A and B.

39.

X

Y

360°0°

4

-4

y = 4 sin x

y = -3 A B

The graphs with equations y = 4 sin x and y = -3 are shown.Find the x co-ordinates of the points of intersection A and B.

40.

X

Y

360°0°

8

-8

y = 8 sin x

y = -7 A B

The graphs with equations y = 8 sin x and y = -7 are shown.Find the x co-ordinates of the points of intersection A and B.

41.

X

Y

360°0°

6

-6y = -5

y = 6 cos x

A B

The graphs with equations y = 6 cos x and y = -5 are shown.Find the x co-ordinates of the points of intersection A and B.

42.

43.

2

X

Y

0

10

360

6y = 7

A By = 6 + 4 sin x°

The graphs with equations y = 6 + 4 sin x° and y = 7 are shown.Find the x co-ordinates of the points of intersection A and B.

44.

X

Y

0

20

12

360

4

y = 15A B

y = 12 + 8 cos x°

The graphs with equations y = 12 + 8 cos x° and y = 15 are shown.Find the x co-ordinates of the points of intersection A and B.

45.

X360°A B

Y

8

-20°

y = 3 + 5 sin x°

The graph with equation y = 3 + 5 sin x° meets the x-axis at the points A and B as shown.Find the x co-ordinates of the points A and B.

48.

2

X

Y

0

18

180

10

y = 16A B

y = 10 + 8 sin 2x°

The graphs with equations y = 10 + 8 sin 2x° and y = 16 are shown.Find the x co-ordinates of the points of intersection A and B.

47.

X

Y

180°0°

3

-3

y = -2

y = 3 cos 2x

A B

The graphs with equations y = 3 cos 2x and y = -2 are shown.Find the x co-ordinates of the points of intersection A and B.

46.

X

Y

180°0°

6

-6

y = 4

A By = 6 sin 2x

The graphs with equations y = 6 sin 2x and y = 4 are shown.Find the x co-ordinates of the points of intersection A and B.

X

Y

9

-9

348-12 168

y = 7A B

y = 9 sin( x + 12)°

The graphs with equations y = 9 sin( x + 12)° and y = 7 are shown.Find the x co-ordinates of the points of intersection A and B.49.

The graphs with equations y = 10 cos( x - 18)° and y = 8 are shown.Find the x co-ordinates of the points of intersection A and B.

-10

X

Y

10

2880 19810818 378

y = 10 cos( x - 18)°

y = 8A B

50.

X

Y

5

-5

37010 190

y = -4

y = 5 sin( x - 10)°

A B

The graphs with equations y = 5 sin( x - 10)° and y = -4 are shown.Find the x co-ordinates of the points of intersection A and B.51.

The graph shows the depth of water in a harbour over a 24 hour period.The depth, D metres, at time t hours after midnight, is given by the formula D = 10 + 6 sin15 t°.

(a) Find the maximum and minimum depths of water in the harbour and the times of day they occur.(b) Find the depth of water in the harbour at 2 pm.(c) To safely leave the harbour a ship needs to have a depth of at least 13 metres of water. Between which two times of day can the ship safely leave the harbour?

min.

t

D

0

max.

12 24

10

The graph has equation D = 10 + 6 sin15 t°.52.

The graph shows the depth of water in a harbour over a 12 hour period.The depth, D metres, at time t hours after midnight, is given by the formula D = 16 - 10 sin 30 t°.

(a) Find the maximum and minimum depths of water in the harbour and the times of day they occur.(b) Find the depth of water in the harbour at 1 am.(c) To safely leave the harbour a ship needs to have a depth of at least 21 metres of water. Between which two times of day can the ship safely leave the harbour?

min.

t

D

0

max.

6 12

16

The graph has equation D = 16 - 10 sin 30 t°.53.

The graph shows the depth of water in a harbour over a 12 hour period.The depth, D metres, at time t hours after midnight, is given by the formula D = 9 + 6 cos( 30 t - 15)°.

(a) Find the maximum and minimum depths of water in the harbour and the times they occur.(b) Find the depth of water in the harbour at midnight.(c) Find the depth of water in the harbour at 6 45 am.(d) To safely leave the harbour a boat needs to have a depth of at least 3.2 metres of water. Between which two times of the morning can the boat not safely leave the harbour?

55.

min.

t

D

0

max.

12

The graph has equation D = 9 + 6 cos( 30 t - 15)°.

The graph shows the depth of water in a harbour over a 36 hour period.The depth, D metres, at time t hours after midnight on Sunday, is given by the formula D = 5 sin(10 t - 15)° + 7.

(a) Find the maximum and minimum depths of water in the harbour and the times they occur.(b) Find the depth of water in the harbour at midnight on Sunday.(c) Find the depth of water in the harbour at 4 30 am on Monday.(d) To safely leave the harbour a boat needs to have a depth of at least 4.5 metres of water. Between which two times can the boat not safely leave the harbour?

54.The graph has equation D = 5 sin(10 t - 15)° + 7.

min.

t

D

0

max.

36Monday Tuesday

Answers1. a = 6 b = 2 2. a = 10 b = 63. a = 3 b = 4 4. a = 8 b = 55. a = 16 b = 3 6. a = 12 b = 27. a = 5 b = 3 8. a = 9 b = 129. a = 11 b = 5 10. a = 18 b = 3

11. a = 4 b = 6 12. a = 14 b = 913. a = 2.5 b = 10 14. a = -7 b = 315. a = 5 b = 0.5 16. a = 3.2 b = 317. a = -6 b = 4 18. a = 8 b = 0.519. a = 3 b = 5 20. a = 4 b = -821. a = 7 b = 10 22. a = 3.6 b = -2023. a = 2 b = 25 24. a = 6.5 b = -1225. a = 20 b = 6 c = 25 26. a = 25 b = 9 c = 2527. a = 10 b = 30 c = 15 28. a = 30 b = 9 c = 4529. a = 17.5 b = 12 c = 20 30. a = 22 b = 40 c = 10

31. max ( 5 , 25) min ( 15 , 5) 32. max ( 0 , 220) , (12 , 220) min ( 6 , 20)33. max ( 6 , 30) min ( 18 , -10) 34. max ( 12 , 35) min ( 32 , 5)35. max ( 3 , 160) min ( 39 , 40) 36. max ( 34.8 , 190) min ( 16.8 , -90)

37. 53.1 and 126.9 38. 56.4 and 123.6

39. 70.5 and 289.5 40. 228.6 and 311.4

41. 241.0 and 299.0 42. 146.4 and 213.6

43. 14.5 and 165.5 44. 68.0 and 292.0

45. 216.9 and 323.1 46. 20.9 and 69.1

47. 65.9 and 114.1 48. 24.3 and 65.7

49. 39.1 and 116.9 50. 54.9 and 341.1

51. 243.1 and 316.9

52. (a) max 6 am ; 16 m min 6 pm ; 4 m (b) 7 m (c) 2 am and 10 am

53. (a) max 9 am ; 26 m min 3 am ; 6 m (b) 11 m (c) 7 am and 11 am

54. (a) max Mon 10.30 am ; 12 m min Tues 4.30 am ; 2m(b) 5.7 m (c) 9.5 m (d) Mon 10.30 pm and Tues 10.30 am

55. (a) max 12.30 am ; 15 m min 6.30 am ; 3 m(b) 14.8 m (c) 3.1 m (d) 6 am and 7 am

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