finding the function describing harmonic motion

[sayyadina]

Amplitude 1/2
Period pi/4
Phase shift -pi

What is the function?


Amplitude = absolute value (1/2)= could be 1/2 or -1/2

b= 2pi/period = 2pi/(pi/4) = 8

c= -b(phase shift) = -8(-pi) = 8pi


So the function could be f(x) = -1/2cos(8x+8pi) right?

Or could it be f(x)= -1/2sin(8x+4pi)?

 

[Romsek]

Amplitude 1/2
Period pi/4
Phase shift -pi

What is the function?

Amplitude = absolute value (1/2)= could be 1/2 or -1/2

b= 2pi/period = 2pi/(pi/4) = 8

c= -b(phase shift) = -8(-pi) = 8pi


So the function could be f(x) = -1/2cos(8x+8pi) right?

Or could it be f(x)= -1/2sin(8x+4pi)?

Amplitudes are always positive.
The phase shift isn't multiplied by b, it's just added.


a general form of the sinosoid with the parameters you have is

ACos((2pi/Period)x + (phase shift))

Note that 1/Period is called the frequency and is usually written as f or \(\displaystyle \nu\) and (2pi f) is usually denoted as \(\displaystyle \omega\)
Phase shift is usually denoted as \(\displaystyle \phi\)

with these you get the simple form of

A Cos(2pi f x + \(\displaystyle \phi\)) or A Cos(\(\displaystyle \omega\) x + \(\displaystyle \phi\))

So back to your problem, using all the above you'd end up with
A = 1/2, \(\displaystyle \omega\) = 2pi/(pi/4) = 8 (as you noted), and \(\displaystyle \phi\) = -pi so you end up with

1/2 Cos(8x - pi)