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Graphs of Sine and Cosine Functions
You’ll need graph paper
4.5
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On Graph paper use a radius of 7in to represent the radius of your unit circle.
Then give both the fractional and decimal value of your trig function for each value of theta on the unit circle. (Do not include the last value of theta for your quadrant) i.e. 90, 180, 270, 360
2
Unit Circle Activity
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Groups 1-4 – Sine Groups 5-8 – Cosine Groups 9-12 – Tangent
Group 4n+1 – Q1 Group 4n+2 – Q2 Group 4n+3 – Q3 Group 4n+4 – Q4
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12 Groups
Ex: Group 7:7 = 4(1)+3
Cosine Q3
n is a whole #
Only need one graph
per group
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Label with “as increases, [trigfunction( )] increases/decreases.”
Ex for sine of Q1: “as increases, increases.”
4
sin
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Graph the sine and cosine functions Graph the sine function on a new piece of graph
paper
Label your x-axis in radians in multiples of . Use 1 square for each measure
On your y-axis label . Count 1 square as ¼.
Then graph the cosine function a separate
axis. (use the same labels)
6
2,2
3,1,
2
1,0
2
6to
Everyone needs to
do their own!
Everyone is starting
with sine
13 squares on each side of y-axis
8 squares on each side of x-axis
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Graph the csc and sec functions
Graph csc with the sin graph and sec with the cos graph
Then make a new graph for tan and cot Label your x-axis in radians in multiples of
7
12
6
5
4
3
3
2
2346
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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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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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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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Amplitude
Defined as the absolute value of maximum or minimum of the function
Try graphingy = 2 sinx What is the amplitude
For y = a cos x or y = a sin x The amplitude is |a| Do we need to worry about the amplitude for the other
trig functinos?
amplitude = 1amplitude = 1 y=sinxy=sinx
2
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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 cot The period is π
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Period of a Trig Function
What happens for ? Try graphing y = sin 2x
What is the period? What about y = sin 3x
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
2
2
3
2
4
2/1
2
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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 for
y = tan (1/3 x)
• Graph it and verify your prediction
• Predict the period fory = tan (1/3 x)
• Graph it and verify your prediction
3
3/1
Same for cotSame for cot
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Review of Transformations
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1)82(3)( 2 xxf
1)]4(2[3)( 2 xxf
Or reflection over x-axis!
Or reflection over y-axis!
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Review of Transformations
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Sketch the graphDo non-rigid transformation 1st (strech/compress)Then rigid transformations (up/down and left/right)
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Let’s investigate with graphs of trig functions!
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dcxbay )](sin[
Desmos.com
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Practice time
Graph
a.
b.
19
12cos2 xy
2sin3
xy
Sketch each transformation of the graph
Sketch between 0 and 2pi while doing transformations
Does the make a difference?
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H Dub
4-5 Pg 328 #1-25odd, 35-51EOO
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Graph the sine and cosine functions
Regraph the sine and cosine functions on two separate axis
Label your graph in radians
On your y-axis labelLeave room above and below!
On your x-axis label multiples of
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1,2
1,0
360,270,225,180,135,90,45,0
2
6to