1. © T Madas

2. The graphs of
sinx, cosx & tanx
© T Madas

3. y = sin x x 15 30 45 60 75 90
105
120
135
150
165
180 y
0.26
0.5
0.71
0.87
0.97 1
0.97
0.87
0.71
0.5
0.26 x
195
210
225
240
255
270
285
300
315
330
345
360 y
-0.26
-0.5
-0.71
-0.87
-0.97 -1
-0.97
-0.87
-0.71
-0.5
-0.26
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5
© T Madas

4. y = cos x x 15 30 45 60 75 90
105
120
135
150
165
180 y 1
0.97
0.87
0.71
0.5
0.26
-0.26
-0.5
-0.71
-0.87
-0.97 -1 x
195
210
225
240
255
270
285
300
315
330
345
360 y
-0.97
-0.87
-0.71
-0.5
-0.26
0.26
0.5
0.71
0.87
0.97 1
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5
© T Madas

5. y = tan x 7 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 -7
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° x 15 30 45 60 75 90
105
120
135
150
165
180 y
0.27
0.58 1
1.73
3.73 
-3.73
-1.73 -1
-0.58
-0.27 x
195
210
225
240
255
270
285
300
315
330
345
360
© T Madas y
0.27
0.58 1
1.73
3.73 
-3.73
-1.73 -1
-0.58
-0.27

6. y = tan x asymptotes 7 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 -7 x 80 85 95 100
260
265
275
280 y
5.67
11.43
-11.43
-5.67
5.67
11.43
-11.43
-5.67
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360°
asymptotes
© T Madas

7. Plotting the trigonometric functions in one graph8 6 4 2 -2 -4 -6 -8
-720
-540
-360
-180
180
360
540
720
© T Madas

8. The graphs of
sinx + c
cosx + c
© T Madas

9. y = sin x + 1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 x y x y 15 30 45 60 75 9015 30 45 60 75 90
105
120
135
150
165
180 y 1
1.26
1.5
1.71
1.87
1.97 2
1.97
1.87
1.71
1.5
1.26 1 x
195
210
225
240
255
270
285
300
315
330
345
360 y
0.74
0.5
0.29
0.13
0.03
0.03
0.13
0.29
0.5
0.74 1
1.5 2 1
-0.5
0.5 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360°
© T Madas

10. What is the equation of the purple curve?
What is the equation of the green curve? 1 2 -1 -2
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360°
© T Madas

11. Plotting sinx + c for different values of c8 6 4 2 -2 -4 -6 -8
-720
-540
-360
-180
180
360
540
720
Sinx + 4
sinx
Sinx – 8
© T Madas

12. The graphs of
a sinx
a cosx
© T Madas

13. y = 2sin x x 15 30 45 60 75 90
105
120
135
150
165
180 y
0.52 1
1.41
1.73
1.93 2
1.93
1.73
1.41 1
0.52 x
195
210
225
240
255
270
285
300
315
330
345
360 y
-0.52 -1
-1.41
-1.73
-1.93 -2
-1.93
-1.73
-1.41 -1
-0.52 1 2 -1 -2
y - stretch
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360°
© T Madas

14. What does y = 2cosx look like?
What is the equation of the green curve? 1 2 -1 -2
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360°
© T Madas

15. Plotting a sinx for a = 1, 2, 3, 4, 5 in one graph8 6 4 2 -2 -4 -6 -8
-720
-540
-360
-180
180
360
540
720
sinx
2sinx
3sinx
4sinx
5sinx
© T Madas

16. Plotting a cosx for a = 1, 2, 3, 4, 5 in one graph8 6 4 2 -2 -4 -6 -8
-720
-540
-360
-180
180
360
540
720
5cosx
4cosx
3cosx
2cosx
cosx
© T Madas

17. The graphs of
-a sinx
-a cosx
-a tanx
© T Madas

18. y = -sin x x 15 30 45 60 75 90
105
120
135
150
165 y
-0.26
-0.5
-0.71
-0.87
-0.97 -1
-0.97
-0.87
-0.71
-0.5
-0.26 x
180
195
210
225
240
255
270
285
300
315
330
345
360 y
0.26
0.5
0.71
0.87
0.97 1
0.97
0.87
0.71
0.5
0.26
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5
© T Madas

19. What does y = -2cosx look like?1 2 -1 -2
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360°
© T Madas

20. 7 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 -7
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360°
© T Madas

21. Plotting a sinx for -3 ≤ a ≤ 3 in one graph8 6 4 2 -2 -4 -6 -8
-720
-540
-360
-180
180
360
540
720
Work out each equation
© T Madas

22. Plotting a cosx for -3 ≤ a ≤ 3 in one graph8 6 4 2 -2 -4 -6 -8
-720
-540
-360
-180
180
360
540
720
Work out each equation
© T Madas

23. The graphs of
sin(ax )
cos(ax )
© T Madas

24. y = sin (2x) x - squash x y x y 15 30 45 60 75 90 105 120 135 150 16515 30 45 60 75 90
105
120
135
150
165
180 y
0.5
0.87 1
0.87
0.5
-0.5
-0.87 -1
-0.87
-0.5 x
195
210
225
240
255
270
285
300
315
330
345
360 y
0.5
0.87 1
0.87
0.5
-0.5
-0.87 -1
-0.87
-0.5
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5
x - squash
© T Madas

25. y = sin (2x) (3x) 1.5 1 0.5 -0.5 -1 180° 210° 240° 270° 300° 330° 30°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5
© T Madas

26. Plotting sin(ax) for various values of a1 -1
-360
-270
-180
-90 90
180
270
360
Work out each equation
Plotting cos(ax) for various values of a 1 -1
-360
-270
-180
-90 90
180
270
360
© T Madas

27. The graphs of
sin(x + a)
cos(x + a)
© T Madas

28. y = sin(x + 30) x 15 30 45 60 75 90
105
120
135
150
165
180 y
0.5
0.71
0.87
0.97 1
0.97
0.87
0.71
0.5
0.26
-0.26
-0.5 x
195
210
225
240
255
270
285
300
315
330
345
360 y
-0.71
-0.87
-0.97 -1
-0.97
-0.87
-0.71
-0.5
-0.26
0.26
0.5
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30
© T Madas

29. y = sin(x + 30) 60 x 15 30 45 60 75 90
105
120
135
150
165
180 y
0.87
0.97 1
0.97
0.87
0.71
0.5
0.26
-0.26
-0.5
-0.71
-0.87 x
195
210
225
240
255
270
285
300
315
330
345
360 y
-0.97 -1
-0.97
-0.87
-0.71
-0.5
-0.26
0.26
0.5
0.71
0.87
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60
© T Madas

30. y = sin(x + 30) 60 – x 15 30 45 60 75 90
105
120
135
150
165
180 y
-0.87
-0.71
-0.5
-0.26
0.26
0.5
0.71
0.87
0.97 1
0.97
0.87 x
195
210
225
240
255
270
285
300
315
330
345
360 y
0.71
0.5
0.26
-0.26
-0.5
-0.71
-0.87
-0.97 -1
-0.97
-0.87
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60
© T Madas

31. What are the equations of these curves?
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5
© T Madas

32. What are the equations of these curves?
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5
© T Madas

33. What are the equations of these curves?
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5
© T Madas

34. What are the equations of these curves?
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5
© T Madas

35. sin(x + 90) = cos x sin x = cos( x – 90)
What are the equations of these curves?
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360° 1
1.5
0.5 -1
-0.5
sin(x + 90) =
cos x
sin x =
cos( x – 90)
© T Madas

36. © T Madas

37. A B C D Equation y = sinx y = cosx y = sin(2x ) y = sin( x ) Letter B
The graphs of four trigonometric functions are shown below.
Write the letter of each graph next to the equation which produced it. A B C D
Equation
y = sinx
y = cosx
y = sin(2x )
y = sin( x ) 1 2
Letter B D A C
© T Madas

38. © T Madas

39. The graph of cos2x is shown below for x between 0° and 360°
Write down a suitable scale on the x and y axis and hence write down the coordinates of the points A, B, C, D, E and F.
Use the graph to find estimates for the values of x for which cos2x = 0.5, for x between 0° and 360°.
360° y x
1.2
y = cos2x D
(180,0) 1
0.8
0.6
0.4
0.2 A
(45,0) C
(135,0)
(305,0) E
(225,0) F
30°
60°
90°
120°
150°
180°
210°
240°
270°
300°
330°
-0.2
-0.4
The value of sine and cosine lies between…
… -1 and 1
-0.6
-0.8 -1 B
(90,-1)
© T Madas
-1.2

40. The graph of cos2x is shown below for x between 0° and 360°
Write down a suitable scale on the x and y axis and hence write down the coordinates of the points A, B, C, D, E and F.
Use the graph to find estimates for the values of x for which cos2x = 0.5, for x between 0° and 360°.
360° y x
1.2
y = cos2x D
(180,0) 1
0.8
0.6
y = ½
0.4
0.2 A
(45,0) C
(135,0) E
(225,0) F
(305,0)
30°
60°
90°
120°
150°
180°
210°
240°
270°
300°
330°
cos2x = 0.5
x = 30°
x = 150°
x = 210°
x = 330°
-0.2
-0.4
-0.6
-0.8 -1 B
(90,-1)
© T Madas
-1.2

41. © T Madas

42. 4 sinx + 1 = 0 c 4 sinx = -1 c sinx = 194° 346°
The graph of y = sinx is drawn below, for x between 0 and 360°
Fill in the scale on the y axis
Use the graph to get estimates for the solution of the equation 4sinx + 1 = 0 for x between 0 and 360°
30°
60°
90°
120°
150°
180°
210°
240°
270°
300°
330°
360° x y
1.2 4
sinx
+ 1
= 0 c 1
0.8 4
sinx = -1 c
0.6
sinx = 1 4
0.4
0.2
194°
346°
-0.2
y = - ¼
-0.4
-0.6
y = sinx
-0.8 -1
-1.2
© T Madas

43. © T Madas

44. The graph of sinx is shown below for x between 0° and 360°
Write down the co ordinates of the points A, B and C.
Sketch the graph of sin2x for x between 0° and 360°. x y
The value of sine and cosine lies between…
… -1 and 1
1.2 A
(90,1) 1
0.8
0.6
0.4
0.2 B
(180,0)
90°
180°
270°
360°
-0.2
y = sinx
-0.4
-0.6
-0.8 -1 C
(270,-1)
-1.2
© T Madas

45. The graph of sinx is shown below for x between 0° and 360°
Write down the co ordinates of the points A, B and C.
Sketch the graph of sin2x for x between 0° and 360°. x y
If y = sinx is y = f(x )
then y = sin2x is…
… y = f(2x )
… i.e. squashed by a factor of 2 in x only.
1.2 A
(90,1) 1
0.8
0.6
0.4
0.2 B
(180,0)
90°
180°
270°
360°
-0.2
y = sin2x
y = sinx
-0.4
-0.6
-0.8 -1 C
(270,-1)
-1.2
© T Madas

46. The graph of sinx is shown below for x between 0° and 360°
Write down the co ordinates of the points A, B and C.
Sketch the graph of sin2x for x between 0° and 360°. x y
If y = sinx is y = f(x )
then y = sin2x is…
… y = f(2x )
… i.e. squashed by a factor of 2 in x only.
1.2 A
(45,1) 1
0.8
0.6
0.4
0.2 B
(90,0) B
(180,0)
90°
180°
270°
360°
-0.2
y = sin2x
-0.4
-0.6
-0.8 -1 C
(135,-1) C
(270,-1)
-1.2
© T Madas

47. © T Madas

48. y = 2 + sin6x y = a + sinbx The equation of a curve is y = a + sinbx.
The curve passes through A (0,2) and B (5,2½).
Find the values of a and b and hence write down the equation of the curve
y = 2 + sin6x
y = a + sinbx
when x = 0, y = 2
2 =
a +
sin(b x 0)
sin 0 = 0
a = 2
when x = 5, y = 2½
2½ =
2 +
sin(b x 5)
½ =
sin(5b)
5b =
sin-1(½)
5b = 30
b = 6
© T Madas

49. © T Madas

50. y = -1 + tan5x y = a + tanbx The equation of a curve is y = a + tanbx.
The curve passes through A (0,-1) and B (9,0).
Find the values of a and b and hence write down the equation of the curve
y = a + tanbx
y = -1 + tan5x
when x = 0, y = -1
-1 =
a +
tan(b x 0)
tan 0 = 0
a = -1
when x = 9, y = 0
0 =
-1 +
tan(b x 9)
1 =
tan(9b)
9b =
tan-1(1)
9b = 45
b = 5
© T Madas

51. © T Madas

52. The graph of y = p + q sin r x is shown below.
Find the values of p, q and r and hence write down the equation of the curve 1 3 2 4 3 2 1 -1 -2 -3
180°
210°
240°
270°
300°
330°
30°
60°
90°
120°
150°
360°
when x = 0, y = 1
1 =
p +
q sin(r x 0)
sin 0 = 0
p = 1
© T Madas

53. © T Madas