Trigonometric Graphs

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

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

1 0.5 30 60 90 120 150 180 210 220 270 300 330 360 -0.5 -1

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

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

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

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

Y = - 8sinx Maximum Value = 8 Minimum Value = -8 “Opposite” to Sin x 8 90 180 270 360 -8 “Opposite” to Sin x Maximum Value = 8 Minimum Value = -8

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

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

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

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

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

Y = sin x Cycle starts again Period of graph is 3600 1 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

Y = sin 2x Period of graph is 1800 90 180 270 360 -1 Period of graph is 1800 There are 2 cycles between 00 and 3600

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

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

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 2 cycles Min -8 Period = 360 ÷ 2 = 1800 8 Y = 8sin 2x 90 180 270 360 -8

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

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

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

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

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

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

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