Question

Urgent Help! 100 points to whoever is willing ^-^

Noise-canceling headphones have microphones to detect the ambient, or background, noise. They interpret those noises as sinusoidal functions. To cancel out that noise, the headphones create their own sinusoidal functions that mimic the incoming noise, but it changes them in one of two ways.
1. The mimic function is the negative of the noise's function.
2. The mimic function is the noise function shifted one-half period.

The headphones then play the noise function together with the mimic function, which cancels the noise.

Instructions
• Find the frequency of any musical note in hertz (Hz).
• Use the frequency to write f(x), the sine function for the note. For example,
`A_4` has a frequency of 440 Hz. In radians, we describe this note as y = sin(440(2πx)) or y − sin(880πx)
• Graph the sine function for the chosen note.
• Use one of the two methods listed above to write g(x), the mimic function that cancels that note's sound. Graph that function.
• Write a third function, h(x), that is the sum of f(x) and g(x). Graph it.
• Use your three graphs to explain why g(x) cancels out f(x).

Answer 1

Let's assume we want to cancel out a musical note with a frequency of f Hz. We can write the sine function for this note as:

f(x) = sin(2πfx)

To cancel out this note, we need to create a mimic function g(x) that is either the negative of f(x) or shifted one-half period. Let's consider the two cases separately:

1. Mimic function is the negative of f(x):

In this case, the mimic function g(x) is simply the negative of f(x):

g(x) = - sin(2πfx)

We can graph f(x) and g(x) together to see how they cancel each other out:

Graph of f(x) and g(x) when g(x) is the negative of f(x):

The blue graph represents f(x), while the red graph represents g(x). As we can see, the two functions are equal in magnitude but opposite in sign. When we add them together, we get:

h(x) = f(x) + g(x) = sin(2πfx) - sin(2πfx) = 0

This means that the sound of the musical note is cancelled out completely.

2. Mimic function is the noise function shifted one-half period:

In this case, the mimic function g(x) is the sine function with the same frequency as f(x), but shifted one-half period:

g(x) = sin(2πfx + π)

We can graph f(x) and g(x) together to see how they cancel each other out:

Graph of f(x) and g(x) when g(x) is the noise function shifted one-half period:

The blue graph represents f(x), while the red graph represents g(x). As we can see, the two functions are equal in magnitude but out of phase by one-half period. When we add them together, we get:

h(x) = f(x) + g(x) = sin(2πfx) + sin(2πfx + π)

Using trigonometric identities, we can simplify this expression to:

h(x) = 2cos(π/2)sin(2πfx + π/2)

h(x) = 2cos(π/2)cos(2πfx) (since sin(2πfx + π/2) = cos(2πfx))

h(x) = 2sin(2πfx)

This means that the sound of the musical note is cancelled out completely.

In summary, the mimic function g(x) cancels out the sound of the musical note by either being the negative of the noise function or being the noise function shifted one-half period. In both cases, the mimic function has the same frequency as the noise function, but is equal in magnitude and opposite in sign or out of phase by one-half period. When we add the noise function and the mimic function together, they cancel each other out and we are left with no sound.