The Wave Function
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Transcript of The Wave Function
![Page 1: The Wave Function](https://reader035.fdocuments.in/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/1.jpg)
The Wave Function
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What is to be learned?
• How the wave function tactic sorts out functions containing sine and cosine
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Previously
Max value of 5sinx is
Min value of 5sinx is5-5
5
-5
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How about 7cosx + 5sinx
Need to rewrite with just sine or cosine
y = 7cosx + 5sinx
change to y = R cos (x – α )
Need to find R and α
angle
sin max at x = 900cos max at x = 00
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y = 7cosx + 5sinx
change to y = R cos (x – α )
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y = 7cosx + 5sinx
change to y = R Cos (x – α )
y = 7 cosx + 5 sinxy = R cosx cosα + R sinx sinα
equating coefficients
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y = 7cosx + 5sinx
change to y = R Cos (x – α )
y = 7 cosx + 5 sinxy = R cosx cosα + R sinx sinα
equating coefficients R cosα= 7
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y = 7cosx + 5sinx
change to y = R Cos (x – α )
y = 7 cosx + 5 sinxy = R cosx cosα + R sinx sinα
equating coefficients R cosα= 7 R sinα = 5
Need to find R and α
sin2x + cos2x = 1R2sin2x + R2cos2x = R2(sin2x + cos2x) = R2
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y = 7cosx + 5sinx
change to y = R Cos (x – α )
y = 7 cosx + 5 sinxy = R cosx cosα + R sinx sinα
equating coefficients R cosα= 7 R sinα = 5
Need to find R and α
R2 = 72 + 52 Sinx
Cosx= Tanx
R
R= √74
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y = 7cosx + 5sinx
change to y = R Cos (x – α )
y = 7 cosx + 5 sinxy = R cosx cosα + R sinx sinα
equating coefficients R cosα= 7 R sinα = 5
Need to find R and α
R2 = 72 + 52 Tan α = 5 7
= 0.714
Tan-1(0.714) = 35.50
or 180 + 35.50
i , iv i , ii
√= √74
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7cosx + 5sinx
= √74 cos(x - 35.50)
Max = √74
Min = - √74
Phase Angle 35.50
Graph moves 35.50 to the right
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The Wave Function
Rewriting functions containing sine and cosine in form
R cos( x – α )
Expand using cos (A – B)
Equate Coefficients
R2 = (R cos α)2 + (R sin α)2
Tan α = R sin α
or similar!
(formula sheet)
R cos αThere can be only one α
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y = 4cosx – 5sinx
change to y = R Cos (x – α )
y = R cosx cosα + R sinx sinαequating coefficients R cosα= 4 R sinα = -5
R2 = 42 + (-5)2 Tan α = -5 4
= -1.25
Tan-1(1.25) = 51.30
360 – 51.30 = 308.70
i , iv iii , iv
iv= √41
Min = - √41Max = √41
Becomes y = √41cos(x – 308.70)
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5cosx – 7sinx
change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα
- 7sinx + 5cosx Equating Coefficients
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5cosx – 7sinx
change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα
- 7sinx + 5cosx Equating Coefficients
Rcos α = -7
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5cosx – 7sinx
change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα
- 7sinx + 5cosx Equating Coefficients
Rcos α = -7 Rsin α = 5–
Rsin α = -5
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Remindersy = sinx y = cosx
Max at x = 900
Min at x = 2700
Max at x = 00
and 3600
Min at x = 1800
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Max Values
Max value of
4sin(x - 30)0
Max value = 4
sinx has max when x = 900
so 4sin(x - 30)0 has max when x - 30 = 90
x = 120
Want this to equal 900
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Min Values
Min value of
8cos(x - 30)0
Min value = -8
cosx has min when x = 1800
so 8cos(x - 30)0 has min when x - 30 = 180
x = 210
Want this to equal 1800
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Uses of the Wave Function
Gets max and min values.
Helps us sketch the graph
and
Good format to solve Trig Equations
May not tell you to use wave function
- look for mix of sin and cos
If you are not told which expansion to use – you get to choose!
Rcos(x – α) – very popular!
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Solve 4cosx – 5sinx = 4
Change to
√41cos(x – 308.70) = 4
Then √41cos A = 4, where A = x – 308.70
cos A = 4/√41
etc.
form Rcos(x – α)