Unit Circle¶
Recall (doc/postdoc). Not a first course.
Job: identify \(\mathbb{R}/2\pi\mathbb{Z}\simeq S^1\); standard angles as roots of unity coordinates.
Core¶
\[
S^1=\{(x,y):x^2+y^2=1\},\qquad
(\cos\theta,\sin\theta)=\text{point at directed angle }\theta
\]
| \(\theta\) | \(0\) | \(\pi/6\) | \(\pi/4\) | \(\pi/3\) | \(\pi/2\) |
|---|---|---|---|---|---|
| \(\cos\) | \(1\) | \(\sqrt3/2\) | \(\sqrt2/2\) | \(1/2\) | \(0\) |
| \(\sin\) | \(0\) | \(1/2\) | \(\sqrt2/2\) | \(\sqrt3/2\) | \(1\) |
Coterminal: \(\theta\sim\theta+2\pi k\). Reference \(\theta'\) in QI with same \(|\sin|,|\cos|\) up to signs.
Special: \(\cos(\theta+2\pi)=\cos\theta\), \(\sin(\theta+\pi)=-\sin\theta\); reflection symmetries of \(D_4\) / \(D_\infty\) action.
Complex: \(e^{i\theta}\in U(1)\); multiplication = angle add.
Used later for¶
- Identities: addition via rotation composition
- Fourier: orthonormal \(\{e^{in\theta}\}\)
- Lie groups: \(U(1)\), \(SO(2)\)
- ODE: \(y''+y=0\) solutions on the circle
Check¶
Is \(\theta\mapsto(\cos\theta,\sin\theta)\) injective on \([0,2\pi)\)?
Yes — bijective onto \(S^1\). On \(\mathbb{R}\) it is \(2\pi\)-periodic, not injective.