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 is5-5
5
-5
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
y = 7cosx + 5sinx
change to y = R cos (x – α )
y = 7cosx + 5sinx
change to y = R Cos (x – α )
y = 7 cosx + 5 sinxy = R cosx cosα + R sinx sinα
equating coefficients
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
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
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
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
7cosx + 5sinx
= √74 cos(x - 35.50)
Max = √74
Min = - √74
Phase Angle 35.50
Graph moves 35.50 to the right
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
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)
5cosx – 7sinx
change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα
- 7sinx + 5cosx Equating Coefficients
5cosx – 7sinx
change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα
- 7sinx + 5cosx Equating Coefficients
Rcos α = -7
5cosx – 7sinx
change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα
- 7sinx + 5cosx Equating Coefficients
Rcos α = -7 Rsin α = 5–
Rsin α = -5
Remindersy = sinx y = cosx
Max at x = 900
Min at x = 2700
Max at x = 00
and 3600
Min at x = 1800
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
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
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!
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)