Question 1013983: 12. A sound wave is modeled with the equation y = 1/4 cos 2pi/3 θ.
a. Find the period. Explain your method.
b. Find the amplitude. Explain your method.
I need help with this! Thank you...
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the basic formula for cosine is:
y = a * (b * (x-c)) + d
a is the amplitude.
b is the frequency.
c is the horizontal displacement.
d is the vertical displacement.
your equation is y = 1/4 * cos (2*pi/3 * x).
i used x instead of theta.
no matter.
just replace my x with your theta and you have your answer.
the amplitude is 1/4.
the frequency is 2 * pi/3.
the horizontal displacement is 0.
the vertical displacement is 0.
if you are dealing in degrees, then the period is equal to 360 divided by the frequency.
if you are dealing in radians, then the period is equal to 2*pi divided by the frequency.
if you know the frequency, you can solve for the period using this formula.
if you know the period, you can solve for the frequency using this formula.
consider f = frequency and p = period
in degrees:
p = 360 / f
solve for f to get f = 360 / p
in radians:
p = 2pi / f
solve for f to get f = 2pi / p
since you're in radians, we'll use the radian formula to get:
p = 2pi/(2pi/3)
this is the same as p = 2pi * 3 / 2pi
the result is that p = 3.
if you graph this equation, your x-axis will be in radians and one full cycle of the cosine wave will take up 3 radians.
here's the graph of your equation.
the normal period is 2pi in radians.
the equation of y = cos(x) is really y = cos(1*x).
1 is the frequency.
p = 2pi/f becomes p = 2pi/1 which becomes p = 2pi.
when the frequency is 1, the period is 2pi.
the following graph of y = cos(x) shows that:
in both cases, the cosine was starts at 1 and ends at 1.
that's because cos(0) = 1 and cos(360) = 1
in radians, cos(0) = 1 and cos(2pi) = 1
also, cos(180) = -1 and cos(pi) = -1
when the period is 2pi, halfway through the period is pi.
when the period is 3, halfway through the period is 1.5.
convert 3 radians to degrees and you get 3 radians * 180 / pi = 171.8873385 degrees.
half of that is equal to 75.84355826 degrees.
convert that to radians and you get 75.84355826 degrees * pi / 180 = .4774648293 * pi radians which is equal to 1.5 radians.
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