the wave function
DESCRIPTION
The Wave Function. What is to be learned?. How the wave function tactic sorts out functions containing sine and cosine. Previously. Max value of 5sinx is Min value of 5sinx is. 5. -5. 5. -5. How about 7cosx + 5sinx. cos max at x = 0 0. sin max at x = 90 0. . - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/1.jpg)
The Wave Function
![Page 2: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/2.jpg)
What is to be learned?
• How the wave function tactic sorts out functions containing sine and cosine
![Page 3: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/3.jpg)
Previously
Max value of 5sinx is
Min value of 5sinx is5-5
5
-5
![Page 4: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/4.jpg)
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
![Page 5: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/5.jpg)
y = 7cosx + 5sinx
change to y = R cos (x – α )
![Page 6: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/6.jpg)
y = 7cosx + 5sinx
change to y = R Cos (x – α )
y = 7 cosx + 5 sinxy = R cosx cosα + R sinx sinα
equating coefficients
![Page 7: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/7.jpg)
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
![Page 8: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/8.jpg)
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
![Page 9: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/9.jpg)
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
![Page 10: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/10.jpg)
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
![Page 11: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/11.jpg)
7cosx + 5sinx
= √74 cos(x - 35.50)
Max = √74
Min = - √74
Phase Angle 35.50
Graph moves 35.50 to the right
![Page 12: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/12.jpg)
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
![Page 13: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/13.jpg)
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)
![Page 14: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/14.jpg)
5cosx – 7sinx
change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα
- 7sinx + 5cosx Equating Coefficients
![Page 15: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/15.jpg)
5cosx – 7sinx
change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα
- 7sinx + 5cosx Equating Coefficients
Rcos α = -7
![Page 16: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/16.jpg)
5cosx – 7sinx
change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα
- 7sinx + 5cosx Equating Coefficients
Rcos α = -7 Rsin α = 5–
Rsin α = -5
![Page 17: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/17.jpg)
Remindersy = sinx y = cosx
Max at x = 900
Min at x = 2700
Max at x = 00
and 3600
Min at x = 1800
![Page 18: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/18.jpg)
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
![Page 19: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/19.jpg)
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
![Page 20: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/20.jpg)
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!
![Page 21: The Wave Function](https://reader035.vdocuments.us/reader035/viewer/2022062408/56814496550346895db13857/html5/thumbnails/21.jpg)
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)