Expert: Steve Holleran - 3/8/2007

Question
Hi Mr.! I need some help..

write an equation of the sine function with each amplitude, period and phase shift.

amp=1/2 , period 3pie/2 , phase shift= -pie/4


wirte an equation of the cosine function with each amplitude, period, and phase shift.

amp= 100, period 630degrees, phase shift= -90degrees

THANKS!

Answer
Hi Kay,

Alright, let's see, what we want is to fit the information into the general sine or cosine equation :

  y = A sin (Bx + C)   or y = A cos (Bx + C)  .

Remember, in these equations, A is the amplitude, The period is always 2pi / B, and the C has to do with the phase shift.  We'll do this last.

For the first one, A = 1/2 ;  to find B, set 2pi/B = 3pi/2, the period you want.  Solving, you'll get B = 4/3.

So now you have :  y = 1/2 sin (4/3 x + C).

To find C, set the quantity in parentheses (called the "argument" of the function) = 0 and solve for x:

You'll find that x = -3C/4.  Now make this equal to the phase shift you want:  -3C/4 = -pi/4, and you'll get that
C = pi/3.

So, your final result is y = 1/2 sin ( 4/3 x + pi/3)


The second one is pretty much the same.  You can leave it in degrees or change to radians.  If you leave it in degrees, then we have, copying the steps above,

A = 100,    360/B = 630, so B = 4/7

and then 4/7 x + C = 0 so x = -7C/4.  Set -7C/4 equal to the phase shift you want:  -7C/4 = -90 and solving you'll get that C = 360/7.

So your equation becomes :  y = 100 cos (4/7 x + 360/7)

or in radians, y = 100 cos (4/7 x + 2pi/7)

I hope this helps you out.

Steve Holleran