The Wave Function

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)