Trigonometry Basics¶
Recall (doc/postdoc). Not a first course.
Job: ratios on the circle / right triangles as coordinates of \(S^1\); SOH-CAH-TOA is the \(\mathbb{R}^2\) special case.
Core¶
Right triangle (acute \(\theta\)): \(\sin\theta=\mathrm{opp}/\mathrm{hyp}\), \(\cos\theta=\mathrm{adj}/\mathrm{hyp}\), \(\tan\theta=\mathrm{opp}/\mathrm{adj}\).
Unit circle: point at angle \(\theta\) (from positive \(x\)-axis) is \((\cos\theta,\sin\theta)\in S^1\).
| Reciprocal | Definition |
|---|---|
| \(\csc\theta\) | \(1/\sin\theta\) |
| \(\sec\theta\) | \(1/\cos\theta\) |
| \(\cot\theta\) | \(1/\tan\theta\) |
Radians: \(\theta\) = arc length on unit circle; full turn \(2\pi\). Degree: \(180^\circ=\pi\) rad.
Signs: quadrants by signs of \(x=\cos\), \(y=\sin\). Reference angle reduces to acute.
Used later for¶
- Unit circle / identities: group law on \(S^1\)
- Complex analysis: \(e^{i\theta}=\cos\theta+i\sin\theta\)
- Fourier analysis: characters of \(S^1\)
- Geometry: law of sines/cosines
Check¶
Why is \(\sin\) not injective on \(\mathbb{R}\), and what covering does that reflect?
Period \(2\pi\) and \(\sin(\pi-x)=\sin x\); \(\mathbb{R}\to S^1\), \(\theta\mapsto e^{i\theta}\) is the universal cover — trig functions factor through the circle.