Graphing Cosine
Graphing cosine involves plotting the wave y = A*cos(Bx + C) + D, which starts at its maximum value and follows a smooth oscillating pattern.
Formula
y = A\cos(Bx + C) + D
Definition
Graphing cosine means drawing the wave shape of $y = \cos(x)$: unlike sine, which starts at zero, cosine starts at its highest point of $1$, comes down, crosses zero, dips to $-1$, and returns to $1$ over a distance of $2\pi$; the graph is the sine graph shifted $\pi/2$ units to the left, since $\cos(x) = \sin(x + \pi/2)$. To graph $y = A\cos(Bx + C) + D$, find amplitude $|A|$, period $2\pi/|B|$, phase shift $-C/B$, and midline $D$, then locate the five key points of one cycle starting at the maximum. Cosine and sine are the real and imaginary parts of the complex exponential, $e^{ix} = \cos(x) + i\sin(x)$, and $\{\cos(nx), \sin(nx)\}$ form an orthonormal basis of $L^2([0, 2\pi])$.
Example
Key points for one cycle of $y = \cos(x)$: $(0, 1)$, $(\pi/2, 0)$, $(\pi, -1)$, $(3\pi/2, 0)$, $(2\pi, 1)$. For $y = -3\cos(2x + \pi/2)$: negative amplitude reflects the graph, so it starts at a minimum, $(-\pi/4, -3)$, crossing up at $(0,0)$, reaching max at $(\pi/4, 3)$. In data analysis, principal component analysis of seasonal temperature data often yields $\cos(2\pi t/12)$ and $\sin(2\pi t/12)$ as its leading components, with the phase of the projection giving the timing of maximum temperature.
Key Insight
The cosine graph is the same shape as the sine graph, just shifted $\pi/2$ units to the left, and negative amplitude ($A < 0$) reflects the graph over the midline, turning maxima into minima, equivalent to adding a phase shift of $\pi$: $-\cos(x) = \cos(x + \pi)$. Cos and sin are eigenfunctions of the derivative operator, $d/dx[\cos(\omega x)] = -\omega\sin(\omega x)$ and $d^2/dx^2[\cos(\omega x)] = -\omega^2\cos(\omega x)$, which is why sinusoids are the natural basis for linear differential equations and why Fourier analysis works for solving PDEs.