Trigonometric Graphs¶
Recall (doc/postdoc). Not a first course.
Job: graphs of \(A\sin(Bx+C)+D\) as affine reparametrizations of periodic signals; amplitude/phase of harmonics.
Core¶
\[
y=A\sin(Bx-C)+D\quad\text{or}\quad A\cos(Bx-C)+D
\]
| Parameter | Meaning |
|---|---|
| $ | A |
| period | $T=2\pi/ |
| phase shift | \(C/B\) (to the right if form \(\sin(B(x-C/B))\)) |
| \(D\) | vertical shift (midline) |
Standard periods: \(\sin,\cos\): \(2\pi\); \(\tan\): \(\pi\) (vertical asymptotes at odd multiples of \(\pi/2\)).
Frequency: \(f=1/T\); angular \(\omega=B\). In signals: \(A\cos(\omega t+\phi)\).
Used later for¶
- Fourier analysis: basis of periods \(2\pi/n\)
- ODE: forced oscillators, resonance
- PDE: separation, standing waves
- Complex analysis: \(\operatorname{Im}e^{i(\omega t+\phi)}\)
Check¶
What is the period of \(\sin x+\sin(\sqrt2\,x)\)?
Not periodic — \(1\) and \(\sqrt2\) incommensurate. Finite trigonometric polynomials with commensurate frequencies are periodic.