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.
in y = a cos (kØ)
a is the amplitude, and the period is 2π/k
so evaluate the period and state the amplitude of your equation.
a. To find the period of the sound wave modeled by the equation y = 1/4 cos (2π/3) θ, we can use the formula for the period of a cosine function: T = 2π/ω, where ω is the angular frequency.
In this case, the angular frequency is 2π/3, so the period is given by T = 2π/(2π/3) = 3.
Therefore, the period of the sound wave is 3. To understand this, note that the general form of the cosine function is y = A cos(Bθ), where A is the amplitude and B determines the frequency. In this case, B is (2π/3), which means the wave completes two full cycles (or oscillations) within the interval of 2π/3 radians. Thus, it takes 3 units of θ to complete one full cycle.
b. To find the amplitude of the sound wave modeled by the equation y = 1/4 cos (2π/3) θ, we look at the coefficient of the cosine function, which in this case is 1/4.
The amplitude of a cosine function, denoted by A, represents the maximum displacement or distance from the centerline or equilibrium position. In this case, the amplitude is 1/4.
Therefore, the amplitude of the sound wave is 1/4. This means that the maximum displacement of the wave from its equilibrium position is 1/4 of a unit.
a. To find the period of the sound wave, we need to understand that the coefficient in front of the angle (θ) in the cosine function represents the number of complete cycles that occur within 2π radians.
In this case, the coefficient in front of θ is 2π/3. This means that, for every 2π/3 radians, the cosine function completes one full cycle.
The period of the wave is defined as the length of one complete cycle. So, to find the period, we need to find the value that satisfies the equation:
2π/3 × period = 2π
To isolate the period, we can divide both sides of the equation by 2π/3:
period = 2π / (2π/3)
Simplifying the expression, we get:
period = 2π × 3 / 2π
period = 3
Therefore, the period of the sound wave is 3.
b. The amplitude of a wave is the maximum displacement from the equilibrium position. In the given equation, the amplitude can be determined from the coefficient in front of the cosine function.
In this case, the coefficient in front of the cosine function is 1/4. This means that the maximum displacement from the equilibrium position is 1/4 of the total range.
The total range of the cosine function is from -1 to 1. Thus, the amplitude is given by:
Amplitude = (1 - (-1)) / 2 * (1/4)
Amplitude = (2) / (8)
Amplitude = 1/4
Therefore, the amplitude of the sound wave is 1/4.