About Sine Curves: Amplitude and Phase Shift
What is a Sine Curve?
A sine curve is a mathematical function that describes a smooth, repetitive oscillation. It is fundamental in trigonometry and has numerous applications in physics, engineering, and signal processing.
Formula
The general form of a sine function with amplitude and phase shift is:
\[ y = A \sin(x - B) \]
Where:
- \(A\) is the amplitude (controls the height of the wave)
- \(B\) is the phase shift (moves the wave left or right)
- \(x\) is the input angle (usually in radians)
- \(y\) is the resulting output
Calculation Steps
- Identify the amplitude (A) and phase shift (B) from the given equation
- For any given x, calculate y using the formula: \(y = A \sin(x - B)\)
- The amplitude A determines the wave's height from the midline
- The phase shift B moves the wave:
- Positive B shifts the wave left
- Negative B shifts the wave right
Example
Let's consider the function \(y = 2 \sin(x - \frac{\pi}{4})\):
- Amplitude (A) = 2
- Phase shift (B) = \(\frac{\pi}{4}\)
- To find y when x = \(\frac{\pi}{2}\):
\[ y = 2 \sin(\frac{\pi}{2} - \frac{\pi}{4}) = 2 \sin(\frac{\pi}{4}) = 2 \cdot \frac{\sqrt{2}}{2} = \sqrt{2} \approx 1.414 \]