Sinusoidal Function

A function with a graph that resembles a sine or cosine curve in the form of:

hckay

hckay

)cos(

)sin(

Where:

2

littlebiga

2

littlebigh

kperiod

2Solve for k:

Calculate c by substituting one set of ordered pairs after replacing A, k, & h.

Big & little: the largest & smallest pieces of the given data.

14.12)sin(61.1

)sin(

6 cty

hcktAy

Next, choose an ordered pair from the data, plug into the equation & solve for c.

10.68 hrs

10.7 hrs

11.98 hrs

12.77 hrs

13.42 hrs

13.75 hrs

13.6 hrs

13.05 hrs

12.3 hrs

11.56 hrs

11.12 hrs

10.53 hrs

a = big minus little, div by 2

h = big plus little div by 2

Period is 12 months (from the data)

Calculate the total hours:

14.12)1sin(61.168.10 6 c

January: t = 1, y = 10.68

)sin(61.146.1 6 c

)sin( 661.146.1 c

161 sin)sin(9068.0sin c

c 61357.1

c

c

6593.1

1357.1 6

14.12)66.1sin(61.1 6 ty

The sinusoidal function representing the data is:

Do the math:

A = Dif of most/least div by 2

h = Sum of most/least div by 2

hcktAy )cos(

45.0)sin(37.0

)cos(

2 cty

hcktAy

t = 0, y = 0.08

c

c

c

c

c

0

coscos1cos

cos1

cos37.037.0

45.0))0(cos(37.008.0

11

2

(A must be negative, since at t=o the value is a minimum, the the graph goes up from there)

However, the phase shift will be different. To avoid a greater phase shift than necessary, use the following rule:

If the value of the function is about zero at x = 0, use sine

If the value of the function is a maximum or minimum at x = 0, use cosine.

The period is 4 seconds

HW: Page 391