Graphing Cosine
TrigonometryGraphing 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 $1$, then comes down, crosses zero, goes to $-1$, and returns to $1$ over a distance of $2\pi$.
Example
Key points for one cycle of $y = \cos(x)$: $(0, 1)$, $(\pi/2, 0)$, $(\pi, -1)$, $(3\pi/2, 0)$, $(2\pi, 1)$. The cosine wave starts at the top.
Key Insight
The cosine graph is the same shape as the sine graph, just shifted $\pi/2$ units to the left. $\cos(x) = \sin(x + \pi/2)$. They are the same wave seen from different starting points.
Definition
To graph $y = A\cos(Bx + C) + D$: find amplitude $|A|$, period $2\pi/|B|$, phase shift $-C/B$, and midline $D$. The five key points of one cycle (starting at the maximum): $x = -C/B$ gives the first max; add $\text{period}/4$ successively for midline-crossing, minimum, midline-crossing, and the next max.
Example
Graph $y = -3\cos(2x + \pi/2)$: $A = -3$ (reflected), $B = 2$, $C = \pi/2$, $D = 0$. Period $= \pi$. Phase shift $= -\pi/4$. Reflection means it starts at a minimum. Start (min): $(-\pi/4, -3)$. Up-cross: $(0, 0)$. Max: $(\pi/4, 3)$. Down-cross: $(\pi/2, 0)$. Min: $(3\pi/4, -3)$.
Key Insight
Negative amplitude ($A < 0$) reflects the graph over the midline, turning maxima into minima and vice versa. This is equivalent to adding a phase shift of $\pi$: $y = -\cos(x) = \cos(x + \pi)$.
Definition
The cosine and sine graphs are the real and imaginary parts of the complex exponential: $e^{ix} = \cos(x) + i\sin(x)$. The relationship $\cos(x) = \sin(x + \pi/2)$ is a $\pi/2$ phase shift, corresponding to multiplication by $e^{i\pi/2} = i$ in the Fourier domain. In functional analysis, $\{\cos(nx), \sin(nx)\}$ for $n = 0, 1, 2, \ldots$ form an orthonormal basis of $L^2([0, 2\pi])$ under the inner product $\langle f, g \rangle = (1/\pi)\int fg$.
Example
In data analysis, principal component analysis (PCA) of seasonal data often yields sine and cosine components as leading principal vectors. Annual temperature data projects onto $\cos(2\pi t/12)$ and $\sin(2\pi t/12)$ as its first two PCA components, with the phase of the projection giving the timing of maximum temperature.
Key Insight
The deep insight is that cos and sin are eigenfunctions of the derivative operator: $d/dx[\cos(\omega x)] = -\omega\sin(\omega x)$ and $d^2/dx^2[\cos(\omega x)] = -\omega^2\cos(\omega x)$. This eigenfunction property is why sinusoids are the natural basis for linear differential equations and why Fourier analysis works for solving PDEs.