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1. Noise canceling headphones work by detecting environmental noise with a microphone and creating sound waves it rads out of phase like the ones shown below.

When added together, the result is the purple line (no sound).
In an airplane, your headphones measure a background noise given by the wave
y = - 60 sin (x - π/7) + 90, measure in decibels. Find the function the speakers would need
to produce to cancel out this background noise using the cosine

1 Noise Canceling Headphones Work By Detecting Environmental Noise With A Microphone And Creating Sound Waves It Rads Out Of Phase Like The Ones Shown Below Whe class=

Sagot :

Answer:  [tex]y= -60\cos\left(\frac{23\pi}{14}-\text{x}\right)+90\\\\[/tex]

Explanation

When we phase shift a sine curve by π radians, we are replacing each copy of x with (x-π). What this does is shift each input π units to the left on the number line, and in turn will shift the curve π units to the right.

Here is what the scratch work looks like:

[tex]y = -60\sin\left(\text{x} - \frac{\pi}{7}\right)+90\\\\y = -60\sin\left((\text{x}-\pi) - \frac{\pi}{7}\right)+90\\\\y = -60\sin\left(\text{x}-\frac{7\pi}{7} - \frac{\pi}{7}\right)+90\\\\y = -60\sin\left(\text{x}+\frac{-7\pi-\pi}{7}\right)+90\\\\y = -60\sin\left(\text{x}-\frac{8\pi}{7}\right)+90\\\\[/tex]

Then your teacher requires you to use a cosine function instead of a sine function. We'll use the identity that [tex]\sin(\text{x}) = \cos\left(\frac{\pi}{2}-\text{x})[/tex] to allow us to convert from sine to cosine.

So,

[tex]\sin(\text{x}) = \cos\left(\frac{\pi}{2}-\text{x}\right)\\\\\sin\left(\text{x}-\frac{8\pi}{7}\right) = \cos\left(\frac{\pi}{2}-\left(\text{x}-\frac{8\pi}{7}\right)\right)\\\\\sin\left(\text{x}-\frac{8\pi}{7}\right) = \cos\left(\frac{\pi}{2}-\text{x}+\frac{8\pi}{7}\right)\\\\\sin\left(\text{x}-\frac{8\pi}{7}\right) = \cos\left(\frac{23\pi}{14}-\text{x}\right)\\\\-60\sin\left(\text{x}-\frac{8\pi}{7}\right)+90 = -60\cos\left(\frac{23\pi}{14}-\text{x}\right)+90\\\\[/tex]

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This is how we arrive at the answer [tex]y = -60\cos\left(\frac{23\pi}{14}-\text{x}\right)+90\\\\[/tex]

I used GeoGebra to confirm that the answer is correct. Desmos is another graphing tool that you could use.

Keep in mind that noise canceling headphones work best when the background noise is consistent and steady. Example: The sound of a lawn mower. For sounds that aren't as predictable/steady, or sudden sounds, the headphones won't work as effectively. This is because the sound's waveform keeps changing, which makes it hard to nail down its opposite (i.e. out of phase) waveform so the two cancel out.

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