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Graphs of Sine and Cosine Functions
Lesson 2.5
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2
Ordered Pairs
Consider the values for x and y in the table to the right
Note Period = 2π Maximum y values Minimum y values
x sin(x) cos(x)
-3.1416 0.0000 -1.0000
-2.6180 -0.5000 -0.8660
-2.0944 -0.8660 -0.5000
-1.5708 -1.0000 0.0000
-1.0472 -0.8660 0.5000
-0.5236 -0.5000 0.8660
0.0000 0.0000 1.0000
0.5236 0.5000 0.8660
1.0472 0.8660 0.5000
1.5708 1.0000 0.0000
2.0944 0.8660 -0.5000
2.6180 0.5000 -0.8660
3.1416 0.0000 -1.0000
3.6652 -0.5000 -0.8660
4.1888 -0.8660 -0.5000
4.7124 -1.0000 0.0000
5.2360 -0.8660 0.5000
5.7596 -0.5000 0.8660
6.2832 0.0000 1.0000
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3
Graphing the Ordered Pairs
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-6.28 -3.14 0.00 3.14 6.28 9.42
sin(x)
cos(x)
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-6.28 -3.14 0.00 3.14 6.28 9.42
sin(x)
cos(x)
Period = 2πPeriod = 2π
Maximum and minimum values
Maximum and minimum values
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4
Graphing on Calculator
Go to ♦Y= screen Enter function
Choose F2, zoom 7-Trig
Graph is plotted Tic marks are in
units of π/2Try Web Graphing
Utility
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5
Amplitude
Defined as the absolute value of maximum or minimum of the function
Try graphingy = 2 cos x What is the amplitude
For y = a cos x or y = a sin x The amplitude is |a|
amplitude = 1amplitude = 1
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6
Period of a Trig Function(Recall slide from previous lesson)
The functions repeat themselves The period is the smallest value, p for
which f(x) = f(x + p)
For sin, cos, sec, csc The period is 2π
For tan and ctn The period is π
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7
Period of a Trig Function
What happens for ? Try graphing y = sin 3x
What is the period?
Try y = cos 0.5x What is the period?
For
Period =
siny b x
siny b x
2
b
Same for cos, sec, cscSame for cos, sec, csc
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8
Period of a Trig Function
For tangent Note amplitude
is without bound Period is π
For
Period =
tany xtany x
tany b x
b
• Predict the period fory = tan (1/3 x)
• Graph it and verify your prediction
• Predict the period fory = tan (1/3 x)
• Graph it and verify your prediction
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9
Assignment
Lesson 2.5 Page 177 Exercises 1 – 61 EOO
also 63