The Wave Function

1. The Wave Function

2. What is to be learned? • How the wave function tactic sorts out functions containing sine and cosine

3. Previously Max value of 5sinx is Min value of 5sinx is 5 -5 5 -5

4. How about 7cosx + 5sinx cos max at x = 00 sin max at x = 900  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

5. y = 7cosx + 5sinx change to y = R cos (x – α )

6. y = R cosx cosα + R sinx sinα y = 7cosx + 5sinx change to y = R Cos (x – α ) y = 7 cosx + 5 sinx equating coefficients

7. y = R cosx cosα + R sinx sinα y = 7cosx + 5sinx change to y = R Cos (x – α ) y = 7 cosx + 5 sinx equating coefficients R cosα = 7

8. y = R cosx cosα + R sinx sinα y = 7cosx + 5sinx change to y = R Cos (x – α ) y = 7 cosx + 5 sinx equating coefficients R cosα = 7 R sinα = 5 Need to find R and α sin2x+ cos2x= 1 R2sin2x+ R2cos2x= R2(sin2x + cos2x) = R2

9. y = R cosx cosα + R sinx sinα y = 7cosx + 5sinx change to y = R Cos (x – α ) y = 7 cosx + 5 sinx equating coefficients R cosα = 7 R sinα = 5 Need to find R and α R2 = 72 + 52 R Sinx = Tanx R Cosx = √74

10. y = R cosx cosα + R sinx sinα y = 7cosx + 5sinx change to y = R Cos (x – α ) y = 7 cosx + 5 sinx equating coefficients i , ii i , iv R cosα = 7 R sinα = 5 Need to find R and α 5 R2 = 72 + 52 Tan α = = 0.714 7 = √74 √ Tan-1(0.714) = 35.50 or 180 + 35.50

11. 7cosx + 5sinx = √74 cos(x - 35.50) Max = √74 Min = - √74 Phase Angle 35.50 Graph moves 35.50 to the right

12. 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) There can be only one α R cos α

13. y = R cosx cosα + R sinx sinα y = 4cosx – 5sinx change to y = R Cos (x – α ) equating coefficients iii , iv i , iv R cosα = 4 R sinα = -5 -5 R2 = 42 + (-5)2 Tan α = = -1.25 4 = √41 Tan-1(1.25) = 51.30 iv 360 – 51.30 = 308.70 Becomes y = √41cos(x – 308.70) Max = √41 Min = - √41

14. 5cosx – 7sinx change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα - 7sinx + 5cosx Equating Coefficients

15. 5cosx – 7sinx change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα - 7sinx + 5cosx Equating Coefficients Rcos α = -7

16. 5cosx – 7sinx change to Rsin(x – α ) = Rsinx cosα – Rcosx sinα - 7sinx + 5cosx Equating Coefficients – Rsin α = 5 Rcos α = -7 Rsin α = -5

17. Reminders y = sinx y = cosx Max at x = 900 Max at x = 00 and 3600 Min at x = 2700 Min at x = 1800

18. Max Values Want this to equal 900 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

19. Min Values Want this to equal 1800 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

20. 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!

21. form Rcos(x – α) 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.