1. Trigonometric Graphs

2. What is to be learned?
How to draw and identify graphs with sine and cosine

3. Y = sinx x 30 60 90
120
150
180
210
240
270
300
330
360
Sin x
0.5
0.87 1
0.87
0.5
-0.5
-0.87 -1
-0.87
-0.5

4. 1
0.5
-0.5 -1

5. Y = sinx 1 90
180
270
360 -1
Maximum Value = 1
Minimum Value = -1

6. Y = cosx 1 90
180
270
360 -1
Maximum Value = 1
Minimum Value = -1

7. Y = 7sinx 7 90
180
270
360 -7
Maximum Value = 7
Minimum Value = -7

8. Y = 4cosx 4 90
180
270
360 -4
Maximum Value = 4
Minimum Value = -4

9. Y = - 8sinx Maximum Value = 8 Minimum Value = -8 “Opposite” to Sin x 890
180
270
360 -8
“Opposite” to Sin x
Maximum Value = 8
Minimum Value = -8

10. Trigonometric Graphs Vital to know the basic shape of sin and cos
The same rules apply to each

11. Y = sinx 1 90
180
270
360 -1
Maximum Value = 1
Minimum Value = -1

12. Y = cosx 1 90
180
270
360 -1
Maximum Value = 1
Minimum Value = -1

13. Type y = A Sinx
If there is a number in front, the graph is the same basic shape, but the limits change
y = 11 sinx 11 90
180
270
360
-11
Max Value = 11
Min Value = -11

14. Y = -9sinx 9 90
180
270
360 -9
“Opposite” to Sin x

15. Y = sin x Cycle starts again Period of graph is 36001 90
180
270
360
450
540 -1
Cycle starts again
Period of graph is 3600
Between 00 and 3600 there is 1 cycle
Also applies to Y = cos x

16. Y = sin 2x Period of graph is 180090
180
270
360 -1
Period of graph is 1800
There are 2 cycles between 00 and 3600

17. Combining these rules Draw y = 6sin2x Y = 6sin 2x Max 6 2 cycles
Min -6
Period = 360 ÷ 2 = 1800 6
Y = 6sin 2x 90
180
270
360 -6

18. Recognising Graph Y = 8cos4x Max 8 4 cycles Cosine Min -8 8 90 180 27090
180
270
360 -8

19. Type y = sin bx Number in front of x tells how many “cycles” there are
y = Sin 3x has 3 cycles
Length of each cycle is called the period.
Period of y = sinx is 3600
Period of y = sin3x
= 360 ÷ 3 = 1200
(up to 3600)

20. Combining our two rules Draw y = 8sin2x
Max 8
2 cycles
Min -8
Period = 360 ÷ 2 = 1800 8
Y = 8sin 2x 90
180
270
360 -8

21. Changing the Scale Nice for Drawing Graphs  y = 6 Sin 3x Cycles?
Period 3
360 ÷ 3 = 1200 6 30 60 90
120 -6

22. Not so nice for recognising graphs 8
150
300
450
600 -8
Period = 600
No of Cycles?
360 ÷ 60 = 6
y = 8 cos 6x

23. Extra Trig Graph Rules
Remember rules for y = (x – 3 )2 + 5
Same rules for trig graphs!
3 units to right
Up 5

24. 450 to right
Y = 4cos (x – 450) 4 90
180
270
360
450 -4
Y = 4cosx

25. always draw normal graph first as a guide
Y = 4cos (x – 450) 4 90
180
270
360
450 -4
Y = 4cosx
always draw normal graph first as a guide

26. 3 2
Y = sinx + 2 1
Y = sinx 90
180
270
360 -1

27. No Maximum (or minimum)
Goes to infinity
What about y = Tanx ??? 90
180
270
360
Cycle complete
Period is 1800
No Maximum (or minimum)