Post on 27-Dec-2015
transcript
© T Madas
© T Madas
The graphs ofsinx, cosx &
tanx
© T Madas
y
x
0
180
0.26
165
0.5
150
0.71
135
0.87
120
0.97
105
1
90
0.970.870.710.50.260
75604530150
0-0.26-0.5-0.71-0.87-0.97-1-0.97-0.87-0.71-0.5-0.26y
360345330315300285270255240225210195x
y = sin x
180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
sinx
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y
x
-1
180
-0.97
165
-0.87
150
-0.71
135
-0.5
120
-0.26
105
0
90
0.260.50.710.870.971
75604530150
10.970.870.710.50.260-0.26-0.5-0.71-0.87-0.97y
360345330315300285270255240225210195x
y = cos x
180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
cosy x=
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
y
x
0
180
-0.27
165
-0.58
150
-1
135
-1.73
120
-3.73
105
90
3.731.7310.580.270
75604530150
0-0.27-0.58-1-1.73-3.733.731.7310.580.27y
360345330315300285270255240225210195x
y = tan x
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-5.67-11.4311.435.67-5.67-11.4311.435.67y
280275265260100958580x
y = tan x
asymptotes
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8
6
4
2
-2
-4
-6
-8
-720 -540 -360 -180 180 360 540 720
cosy x=
tany x=
siny x=
Plotting the trigonometric functions in one graph
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The graphs ofsinx + ccosx + c
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1.5
2
1
-0.5
0.5
y
x
1
180
1.26
165
1.5
150
1.71
135
1.87
120
1.97
105
2
90
1.971.871.711.51.261
75604530150
10.740.50.290.130.0300.030.130.290.50.74y
360345330315300285270255240225210195x
y = sin x + 1
sin 1y x= +
+1
+1
+1+1
+1
+1
+1 +1
+1
+1siny x=
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
2
-1
-2
cos 1y x= -
cosy x=
12
cosy x= +
What is the equation of the purple curve?What is the equation of the green curve?
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8
6
4
2
-2
-4
-6
-8
-720 -540 -360 -180 180 360 540 720
Plotting sinx + c for different values of c
sinx
Sinx + 4
Sinx – 8
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The graphs ofa sinxa cosx
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
2
-1
-2
y
x
0
180
0.52
165
1
150
1.41
135
1.73
120
1.93
105
2
90
1.931.731.4110.520
75604530150
0-0.52-1-1.41-1.73-1.93-2-1.93-1.73-1.41-1-0.52y
360345330315300285270255240225210195x
y = 2sin x
2siny x=
siny x=y -
str
etc
h
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
2
-1
-2
cosy x=
12cosy x=
What does y = 2cosx look like?What is the equation of the green curve?
2cosy x=
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8
6
4
2
-2
-4
-6
-8
-720 -540 -360 -180 180 360 540 720
Plotting a sinx for a = 1, 2, 3, 4, 5 in one graph
sinx2sinx3sinx4sinx5sinx
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8
6
4
2
-2
-4
-6
-8
-720 -540 -360 -180 180 360 540 720
Plotting a cosx for a = 1, 2, 3, 4, 5 in one graph
cosx2cosx3cosx4cosx5cosx
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The graphs of-a sinx-a cosx-a tanx
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y
x
0
360
0.26
345
0.5
330
0.71
315
0.87
300
0.97
285
1
270
0.970.870.710.50.260
255240225210195180
-0.26-0.5-0.71-0.87-0.97-1-0.97-0.87-0.71-0.5-0.260y
1651501351201059075604530150xy = -sin x
180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5 -siny x=
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
2
-1
-2
cosy x=
-2cosy x=
What does y = -2cosx look like?
2cosy x=
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
tany x=
-tany x=
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8
6
4
2
-2
-4
-6
-8
-720 -540 -360 -180 180 360 540 720
Plotting a sinx for -3 ≤ a ≤ 3 in one graph
Work out each equation
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8
6
4
2
-2
-4
-6
-8
-720 -540 -360 -180 180 360 540 720
Plotting a cosx for -3 ≤ a ≤ 3 in one graph
Work out each equation
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The graphs ofsin(ax )cos(ax )
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y
x
0
180
-0.5
165
-0.87
150
-1
135
-0.87
120
-0.5
105
0
90
0.50.8710.870.50
75604530150
0-0.5-0.87-1-0.87-0.500.50.8710.870.5y
360345330315300285270255240225210195x
y = sin (2x)
180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
siny x=
sin(2 )y x=
x - squash
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y = sin (2x)
180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
siny x=
(3x)
sin(2 )y x=
sin(3 )y x=
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Plotting sin(ax) for various values of a1
-1
-360 -270 -180 -90 90 180 270 360
Work out each equation
Plotting cos(ax) for various values of a1
-1
-360 -270 -180 -90 90 180 270 360
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The graphs ofsin(x + a) cos(x + a)
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y
x
-0.5
180
-0.26
165
0
150
0.26
135
0.5
120
0.71
105
0.87
90
0.9710.970.870.710.5
75604530150
0.50.260-0.26-0.5-0.71-0.87-0.97-1-0.97-0.87-0.71y
360345330315300285270255240225210195x
y = sin(x + 30)
180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
siny x=
sin( 30)y x= +
30
30
30
30
3030
30
30
30
30
30
3030
30
30
30
30
30
30
30
30
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y
x
-0.87
180
-0.71
165
-0.5
150
-0.26
135
0
120
0.26
105
0.5
90
0.710.870.9710.970.87
75604530150
0.870.710.50.260-0.26-0.5-0.71-0.87-0.97-1-0.97y
360345330315300285270255240225210195x
y = sin(x + 30)
180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
siny x= sin( 60)y x= +
60
60
60
60
60
6060
60
60
60
60
60
6060
60
60
60
60
60
60
60
60
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y
x
0.87
180
0.97
165
1
150
0.97
135
0.87
120
0.71
105
0.5
90
0.260-0.26-0.5-0.71-0.87
75604530150
-0.87-0.97-1-0.97-0.87-0.71-0.5-0.2600.260.50.71y
360345330315300285270255240225210195x
y = sin(x + 30)
180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
siny x= sin( 60)y x= -
60
60
60
60
6060
60
60
6060
60
60
60
60
60
60
60
60
60
60
60
60
60–
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
siny x=sin( 40)y x= +
What are the equations of these curves?
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
siny x= sin( 60)y x= -
What are the equations of these curves?
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
siny x= sin( 70)y x= +
What are the equations of these curves?
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
siny x= sin( 100)y x= -
What are the equations of these curves?
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180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°
1
1.5
0.5
-1
-0.5
siny x=
sin( 10)y x= +
sin(x + 90) =cos x
cosy x=
sin x =cos( x – 90)
What are the equations of these curves?
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© T Madas
5
4
3
2
1
-1
-2
-3
-4
-5
-600 -500 -400 -300 -200 -100 100 200 300 400 500 600
5
4
3
2
1
-1
-2
-3
-4
-5
-600 -500 -400 -300 -200 -100 100 200 300 400 500 6005
4
3
2
1
-1
-2
-3
-4
-5
-600 -500 -400 -300 -200 -100 100 200 300 400 500 600
5
4
3
2
1
-1
-2
-3
-4
-5
-600 -500 -400 -300 -200 -100 100 200 300 400 500 600
The graphs of four trigonometric functions are shown below.Write the letter of each graph next to the equation which produced it.
A B
C D
CADBLetter
y = sin(2x )y = cosxy = sinxEquation y = sin( x )12
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© T Madas
360°0
y
x
1.2
1
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
-0.8
-1
-1.2
30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330°
The graph of cos2x is shown below for x between 0° and 360°
1. Write down a suitable scale on the x and y axis and hence write down the coordinates of the points A, B, C, D, E and F.
2. Use the graph to find estimates for the values of x for which cos2x = 0.5, for x between 0° and 360°.y = cos2x
The value of sine and cosine lies between…… -1 and 1
A
B
C
D
E F(45,0)
(90,-1)
(135,0)
(180,0)
(225,0) (305,0)
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360°0
y
x
1.2
1
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
-0.8
-1
-1.2
30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330°
The graph of cos2x is shown below for x between 0° and 360°
1. Write down a suitable scale on the x and y axis and hence write down the coordinates of the points A, B, C, D, E and F.
2. Use the graph to find estimates for the values of x for which cos2x = 0.5, for x between 0° and 360°.y = cos2x
A
B
C
D
E F(45,0)
(90,-1)
(135,0)
(180,0)
(225,0) (305,0)
y = ½
cos2x = 0.5
x = 30°x = 150°x = 210°x = 330°
© T Madas
© T Madas
30°0
60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°
x
y1.2
1
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
-0.8
-1
-1.2
y = sinx
The graph of y = sinx is drawn below, for x between 0 and 360°
1. Fill in the scale on the y axis 2. Use the graph to get estimates for the solution of the
equation 4sinx + 1 = 0 for x between 0 and 360°
4 sinx = 0+ 1
4 sinx = -1
sinx = 14
y = - ¼
194° 346°
© T Madas
© T Madas
x
y1.2
1
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
-0.8
-1
-1.2
90° 180° 270° 360°0
The graph of sinx is shown below for x between 0° and 360°
1. Write down the co ordinates of the points A, B and C.
2. Sketch the graph of sin2x for x between 0° and 360°.
y = sinx
A
B
C
The value of sine and cosine lies between…… -1 and 1
(180,0)
(90,1)
(270,-1)
© T Madas
x
y1.2
1
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
-0.8
-1
-1.2
90° 180° 270° 360°0
The graph of sinx is shown below for x between 0° and 360°
1. Write down the co ordinates of the points A, B and C.
2. Sketch the graph of sin2x for x between 0° and 360°.
y = sinx
A
B
C
If y = sinx is y = f(x )then y = sin2x is…… y = f(2x )… i.e. squashed by a factor of 2 in x only.(180,0)
(90,1)
(270,-1)
y = sin2x
© T Madas
x
y1.2
1
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
-0.8
-1
-1.2
90° 180° 270° 360°0
The graph of sinx is shown below for x between 0° and 360°
1. Write down the co ordinates of the points A, B and C.
2. Sketch the graph of sin2x for x between 0° and 360°.
A
B
C
(180,0)
(45,1)
(135,-1) C(270,-1)
If y = sinx is y = f(x )then y = sin2x is…… y = f(2x )… i.e. squashed by a factor of 2 in x only.
y = sin2x
B(90,0)
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© T Madas
The equation of a curve is y = a + sinbx.
The curve passes through A (0,2) and B (5,2½).Find the values of a and b and hence write down the equation of the curve
when x = 0, y = 2
y = a + sinbx
2 = sin(b x 0)a = 2
when x = 5, y = 2½
b = 6
a + sin 0 = 0
2½ = sin(b x 5)2 +½ = sin(5b)
5b = sin-1(½)5b = 30
y = 2 + sin6x
© T Madas
© T Madas
y = a + tanbx
The equation of a curve is y = a + tanbx.
The curve passes through A (0,-1) and B (9,0).Find the values of a and b and hence write down the equation of the curve
when x = 0, y = -1
-1 = tan(b x 0)a = -1
when x = 9, y = 0
b = 5
a + tan 0 = 0
0 = tan(b x 9)-1 +1 = tan(9b)
9b = tan-1(1)9b = 45
y = -1 + tan5x
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© T Madas
4
3
2
1
-1
-2
-3
180° 210° 240° 270° 300° 330°30° 60° 90° 120° 150° 360°
The graph of y = p + q sin r x is shown below.
Find the values of p, q and r and hence write down the equation of the curve
when x = 0, y = 1
1 = q sin(r x 0)p = 1
p + sin 0 = 0
1 3 2
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