What is the formula for a sinusoidal function?
2 Answers
A sinusoidal function can be expressed in the general form:
\[ y = A \sin(B(x - C)) + D \]
or
\[ y = A \cos(B(x - C)) + D \]
where:
- \( A \) is the amplitude (the peak value).
- \( B \) affects the period of the function, with the period calculated as \( \frac{2\pi}{|B|} \).
- \( C \) is the horizontal shift (phase shift).
- \( D \) is the vertical shift.
This formula allows you to describe a wide variety of wave-like behaviors. If you need more details about any part of the formula, feel free to ask!
\[ y = A \sin(B(x - C)) + D \]
or
\[ y = A \cos(B(x - C)) + D \]
where:
- \( A \) is the amplitude (the peak value).
- \( B \) affects the period of the function, with the period calculated as \( \frac{2\pi}{|B|} \).
- \( C \) is the horizontal shift (phase shift).
- \( D \) is the vertical shift.
This formula allows you to describe a wide variety of wave-like behaviors. If you need more details about any part of the formula, feel free to ask!
The general formula for a sinusoidal function can be expressed as:
\[ f(t) = A \sin(2 \pi f t + \phi) \]
or
\[ f(t) = A \cos(2 \pi f t + \phi) \]
where:
- \( A \) is the amplitude of the sinusoid, which represents the peak value of the wave.
- \( f \) is the frequency of the sinusoid, measured in hertz (Hz), which indicates how many cycles the wave completes in one second.
- \( t \) is the time variable.
- \( \phi \) is the phase shift, which determines the horizontal displacement of the wave.
### Key Components:
1. **Amplitude (A):** This determines the maximum value of the sinusoidal function. It represents the distance from the midpoint (or equilibrium) to the peak of the wave.
2. **Frequency (f):** This defines how many complete cycles occur per second. It is inversely related to the period \( T \), where \( T = \frac{1}{f} \).
3. **Phase Shift (φ):** This describes the horizontal shift of the sinusoidal function. A positive phase shift moves the graph to the left, while a negative phase shift moves it to the right.
4. **Angular Frequency (ω):** Sometimes, you might see the formula written with angular frequency \( \omega \), where \( \omega = 2 \pi f \). In this case, the formula would be:
\[ f(t) = A \sin(\omega t + \phi) \]
or
\[ f(t) = A \cos(\omega t + \phi) \]
### Example:
For a sinusoidal function representing a simple AC signal, you might have:
\[ f(t) = 5 \sin(2 \pi \cdot 60 t + \frac{\pi}{4}) \]
In this case:
- The amplitude \( A \) is 5.
- The frequency \( f \) is 60 Hz.
- The phase shift \( \phi \) is \( \frac{\pi}{4} \) radians.
This formula describes a wave with a peak value of 5 units, oscillating at 60 cycles per second, and shifted horizontally by \( \frac{\pi}{4} \) radians.
\[ f(t) = A \sin(2 \pi f t + \phi) \]
or
\[ f(t) = A \cos(2 \pi f t + \phi) \]
where:
- \( A \) is the amplitude of the sinusoid, which represents the peak value of the wave.
- \( f \) is the frequency of the sinusoid, measured in hertz (Hz), which indicates how many cycles the wave completes in one second.
- \( t \) is the time variable.
- \( \phi \) is the phase shift, which determines the horizontal displacement of the wave.
### Key Components:
1. **Amplitude (A):** This determines the maximum value of the sinusoidal function. It represents the distance from the midpoint (or equilibrium) to the peak of the wave.
2. **Frequency (f):** This defines how many complete cycles occur per second. It is inversely related to the period \( T \), where \( T = \frac{1}{f} \).
3. **Phase Shift (φ):** This describes the horizontal shift of the sinusoidal function. A positive phase shift moves the graph to the left, while a negative phase shift moves it to the right.
4. **Angular Frequency (ω):** Sometimes, you might see the formula written with angular frequency \( \omega \), where \( \omega = 2 \pi f \). In this case, the formula would be:
\[ f(t) = A \sin(\omega t + \phi) \]
or
\[ f(t) = A \cos(\omega t + \phi) \]
### Example:
For a sinusoidal function representing a simple AC signal, you might have:
\[ f(t) = 5 \sin(2 \pi \cdot 60 t + \frac{\pi}{4}) \]
In this case:
- The amplitude \( A \) is 5.
- The frequency \( f \) is 60 Hz.
- The phase shift \( \phi \) is \( \frac{\pi}{4} \) radians.
This formula describes a wave with a peak value of 5 units, oscillating at 60 cycles per second, and shifted horizontally by \( \frac{\pi}{4} \) radians.