Transcript Trigonometric Equations

Trigonometric Graphs
What is to be learned?
• How to draw and identify graphs with sine
and cosine
Y = sinx
x
Sin x
0
30
60
90
0
0.5 0.87 1
120 150 180 210 240 270 300 330 360
0.87 0.5
0
-0.5 -0.87 -1 -0.87 -0.5 0
1
0.5
0
30
-0.5
-1
60
90
120
150
180
210
220
270
300
330 360
Y = sinx
1
0
90
180
-1
Maximum Value = 1
Minimum Value = -1
270
360
Y = cosx
1
0
90
180
-1
Maximum Value = 1
Minimum Value = -1
270
360
Y = 7sinx
7
0
90
180
-7
Maximum Value = 7
Minimum Value = -7
270
360
4
Y = 4cosx
0
90
180
-4
Maximum Value = 4
Minimum Value = -4
270
360
Y = - 8sinx
8
0
90
180
-8
“Opposite” to Sin x
Maximum Value = 8
Minimum Value = -8
270
360
Trigonometric Graphs
• Vital to know the basic shape of sin and
cos
• The same rules apply to each
Y = sinx
1
0
90
180
-1
Maximum Value = 1
Minimum Value = -1
270
360
Y = cosx
1
0
90
180
-1
Maximum Value = 1
Minimum Value = -1
270
360
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
0
90
180
-11
Max Value = 11
Min Value = -11
270
360
Y = -9sinx
9
0
90
180
-9
“Opposite” to Sin x
270
360
Y = sin x
1
0
90
180
270
360
450
-1
Cycle starts again
Period of graph is 3600
Between 00 and 3600 there is 1 cycle
Also applies to Y = cos x
540
Y = sin 2x
1
0
90
180
270
360
-1
Period of graph is 1800
There are 2 cycles between 00 and 3600
Combining these rules
Draw y = 6sin2x
Max 6
2 cycles
Min -6
Period = 360 ÷ 2 = 1800
6
Y = 6sin 2x
0
90
-6
180
270
360
Recognising Graph
Y = 8cos4x
Max 8
4 cycles
Cosine
Min -8
8
0
90
-8
180
270
360
Type y = sin bx
Number in front of x tells how many
“cycles” there are (up to 3600)
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
Combining our two rules
Draw y = 8sin2x
Max 8
2 cycles
Min -8
Period = 360 ÷ 2 = 1800
8
Y = 8sin 2x
0
90
-8
180
270
360
Changing the Scale
Nice for Drawing Graphs 
y = 6 Sin 3x
Cycles? 3
Period 360 ÷ 3 = 1200
6
0
-6
30
60
90
120
Not so nice for recognising graphs 
8
0
150
300
450
600
-8
Period = 600
No of Cycles?
y = 8 cos 6x
360 ÷ 60 = 6
Extra Trig Graph Rules
Remember rules for y = (x – 3 )2 + 5
3 units to right
Same rules for trig graphs!
Up 5
Y = 4cos (x –
450)
450 to right
4
0
90
-4
Y = 4cosx
180
270
360
450
Y = 4cos (x – 450)
4
0
90
180
270
360
450
-4
Y = 4cosx
always draw normal
graph first as a guide
3
2
Y = sinx + 2
1
Y = sinx
0
90
-1
180
270
360
Goes to infinity
What about y = Tanx ???
0
90
Period is
1800
180
270
360
Cycle complete
No Maximum (or minimum)