For sine function (sin) and cosine function (cos), their periods are fixed and can be expressed by the following formula:
t = 2π / ω
Where t represents the period, π is pi (approximately equal to 3. 14 159), and ω is the angular frequency of the function (in radians). The relationship between angular frequency and ordinary frequency (in seconds) is ω = 2πf, where f is frequency. Therefore, the periodic formula can also be expressed as:
t = 1 / f
This means that the length of the period is equal to the reciprocal of the frequency.
It should be noted that the periodic formula is suitable for periodic functions, such as sine function and cosine function, where the independent variable is angle or time. For other types of periodic events or phenomena, there may be different periodic calculation methods.
Derivation of periodic test formula
The derivation of the period (t) formula can be based on the properties of sine function or cosine function. We take sine function as an example to deduce.
Sine function is a periodic function, defined as f(x) = A * sin(ωx+φ), where a is the amplitude, ω is the angular frequency and φ is the initial phase.
To derive the periodic formula, it is necessary to find out the characteristics of sine function in a complete period.
Consider the sine function sin(ωx) with a period of 2π. This means that when the independent variable ωx increases by 2π, the function value will be equal to the initial value again.
Therefore, we can get the following relationship:
sin(ωx + 2π) = sin(ωx)
Now, we apply the above relationship to the definition of sine function:
sin(ωx + φ) = sin(ωx)
According to the trigonometric identity sin(A+B) = sinAcosB+cosAsinB, we can expand the above equation:
sin(ωx)cos(φ)+cos(ωx)sin(φ)= sin(ωx)
In order to realize that this equation holds for all x, the coefficients of the corresponding terms must be equal, that is:
cos(φ) = 1
sin(φ) = 0
Since cos(φ) = 1, we can get φ = 0. This means that the initial phase φ is 0.
Since sin(φ) = 0, we can get that sin(0) = 0. This means that when the initial phase of the sine function is 0, its value is 0.
Therefore, we conclude that when ωx increases by a complete period (2π), the value of the sine function will be equal to the initial value of 0 again. In other words, the period of sine function is 2π/ω.
We can express the period as t = 2π/ω, where t is the period and ω is the angular frequency.
This is the derivation of the periodic t formula. For cosine function, similar derivation can be made and the same periodic formula can be obtained.
Common application scenarios of periodic formula (t = 2π/ω)
1. Physics: In physics, many phenomena are periodic, such as the vibration, fluctuation and rotation of objects. The period formula can be used to calculate the period of these periodic events. For example, in simple harmonic vibration, the period formula can be used to calculate the period of vibration.
2. Signal processing and communication: In the field of signal processing and communication, periodic signals are very common. Through the period formula, the period of the signal can be calculated, which helps to analyze and process the signal. For example, in audio signal processing, the periodic formula can be used to determine the periodic characteristics of tones or audio signals.
3. Electrical and electronic engineering: In circuit analysis and electronic engineering, the period formula can be used to calculate the period of AC electrical signals. For sinusoidal AC signals, the period formula is helpful to determine the frequency and period of the signal.
4. Optics: In optics, the period formula can be used to calculate the period of light waves. For example, the electromagnetic wave of visible light can be used to calculate the period length of light wave.
5. Mathematics and engineering calculation: Periodic formula is also widely used in mathematics and engineering calculation. It can be used to calculate the period length of periodic function, thus helping to establish mathematical models and solve engineering problems.
Example of periodic t formula
Example: The vibration frequency of a string is 50 Hz. What is the period of this string?
A: We know that there is the following relationship between frequency f and period t: f =1/t.
Given a frequency f of 50 Hz, we can get 50 = 1/t by substituting it into the formula.
Transforming this equation into the form of period t, we can get: t = 1/50 = 0.02 seconds.
Therefore, the period of this string is 0.02 seconds.
Please note that in the calculation process, it is necessary to ensure the consistency of units, such as converting the unit of frequency from hertz (Hz) to seconds (s) to match the unit of period.