Amplitude
TrigonometryAmplitude is the maximum displacement of a periodic function from its midline, determining the height of peaks and depth of troughs.
Formula
\text{amplitude} = |A| \text{ in } y = A\sin(Bx + C) + D
Definition
Amplitude is how tall a wave is from its middle to its highest point (or from the middle to its lowest point). For the basic sine wave, the amplitude is $1$.
Example
$y = 3\sin(x)$ has amplitude $3$. The wave goes from $-3$ to $+3$. $y = 0.5\cos(x)$ has amplitude $0.5$, a flatter wave that only goes from $-0.5$ to $+0.5$.
Key Insight
Think of amplitude as the volume of a sound wave or the size of an ocean wave. Bigger amplitude means bigger swings up and down.
Definition
For $y = A\sin(Bx + C) + D$ or $y = A\cos(Bx + C) + D$, the amplitude is $|A|$. It is the distance from the midline ($y = D$) to either the maximum or minimum. The graph oscillates between $D - |A|$ and $D + |A|$.
Example
$y = -4\cos(2x) + 1$ has amplitude $|-4| = 4$, midline $y = 1$, max value $= 1 + 4 = 5$, min value $= 1 - 4 = -3$. The negative sign reflects the graph vertically but does not change the amplitude.
Key Insight
Amplitude is always positive (it is the absolute value of $A$). A negative $A$ value means the graph is flipped upside down (reflected over the midline), but the amplitude (the "size" of the wave) is still the positive value.
Definition
For a sinusoidal function $f(x) = A\sin(Bx + C) + D$, amplitude $|A| = (\max f - \min f)/2$. In signal processing, amplitude corresponds to the magnitude of a Fourier coefficient: for $f(x) = \sum A_n \sin(nx + \phi_n)$, each $|A_n|$ is the amplitude of the $n$th harmonic. The total energy of the signal is proportional to $\sum A_n^2$ by Parseval's theorem.
Example
A sound wave with frequency $440$ Hz (A4) and amplitude $0.8$ Pa is represented as $p(t) = 0.8\sin(2\pi \cdot 440t)$. Doubling the amplitude to $1.6$ increases the sound intensity by $4\times$ (since intensity $\sim \text{amplitude}^2$).
Key Insight
In quantum mechanics, the amplitude of a wave function determines the probability density: $|\psi(x)|^2$ gives the probability of finding a particle at position $x$. The Born rule connects the mathematical amplitude to observable probability, making amplitude central to quantum measurement theory.