Trigonometric Graphs

What is to be learned?

• How to draw and identify graphs with sine and cosine

Y = sinx

0 30 60 90 120 150 180 210 240 270 300 330 360x

Sin x 0 10.5 0.870.87 00.5 -0.5-0.5 -0.87-0.87 -1 0

30 60 90 120 150 180 210 220 270 300 330 360

1

0.5

-0.5

-1

0

90 180 270 360

1

0

-1

Y = sinx

Maximum Value = 1

Minimum Value = -1

90 180 270 360

1

0

-1

Y = cosx

Maximum Value = 1

Minimum Value = -1

90 180 270 360

7

0

-7

Y = 7sinx

Maximum Value = 7

Minimum Value = -7

90 180 270 360

4

0

-4

Y = 4cosx

Maximum Value = 4

Minimum Value = -4

90 180 270 360

8

0

-8

Y = - 8sinx

Maximum Value = 8

Minimum Value = -8

“Opposite” to Sin x

Trigonometric Graphs

• Vital to know the basic shape of sin and cos

• The same rules apply to each

90 180 270 360

1

0

-1

Y = sinx

Maximum Value = 1

Minimum Value = -1

90 180 270 360

1

0

-1

Y = cosx

Maximum Value = 1

Minimum Value = -1

90 180 270 360

11

0

-11

Type y = A Sinx

Max Value = 11Min Value = -11

If there is a number in front, the graph is the same basic shape, but the limits change

y = 11 sinx

90 180 270 360

9

0

-9

Y = -9sinx

“Opposite” to Sin x

90 180 270 360

1

0

-1

Y = sin x

540450

Period of graph is 3600

Cycle starts again

Also applies to Y = cos x

Between 00 and 3600 there is 1 cycle

90 180 270 360

1

0

-1

Y = sin 2x

Period of graph is 1800

There are 2 cycles between 00 and 3600

Combining these rules

Draw y = 6sin2x

Max 6

Min -6

2 cycles

Period = 360 ÷ 2 = 1800

90 180 270 360

6

0

-6

Y = 6sin 2x

Recognising Graph

Max 8

Min -8

4 cycles

90 180 270 360

8

0

-8

Y = 8cos4x

Cosine

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)

Combining our two rules

Draw y = 8sin2x

Max 8

Min -8

2 cycles

Period = 360 ÷ 2 = 1800

90 180 270 360

8

0

-8

Y = 8sin 2x

Changing the ScaleNice for Drawing Graphs y = 6 Sin 3x

Cycles?

Period3

360 ÷ 3 = 1200

30 60 90 120

6

0

-6

150 300 450 600

8

0

-8

Not so nice for recognising graphs

Period = 600

No of Cycles? 360 ÷ 60 = 6

y = 8 cos 6x

Remember rules for y = (x – 3 )2 + 5

Same rules for trig graphs!

3 units to right Up 5

Extra Trig Graph Rules

90 180 270 360

4

0

-4

Y = 4cos (x – 450)

450

Y = 4cosx

450 to right

90 180 270 360

4

0

-4

Y = 4cos (x – 450)

450

Y = 4cosx always draw normal graph first as a guide

90 180 270 360

1

0

-1

Y = sinx + 2

Y = sinx

2

3

90 180 270 3600

No Maximum (or minimum)

What about y = Tanx ???Goes to infinity

Cycle completePeriod is 1800