Vertical Lines: Equation $x=c$. Slope is undefined.

General Form: $Ax+By=C$.

Finding Intersections: Substitute $y=ax+b$ into other equation (e.g., circle).

Definition: Angle $A$ has $x$ radians if $\text{Area}(Sector)/\text{Area}(Disc) = x/(2\pi)$.
Conversions: $360^\circ = 2\pi \text{ rad}$, $180^\circ = \pi \text{ rad}$. $1 \text{ rad} = 180/\pi \text{ deg}$.
Arc Length: For a circle of radius $r$, arc length for angle $\theta$ (in radians) is $r\theta$.

Definition: For an angle $A$ with ray passing through $(a,b)$ (not origin), let $r=\sqrt{a^2+b^2}$.
- $\sin A = b/r$.
- $\cos A = a/r$.
Unit Circle: If $r=1$, $(\cos A, \sin A)$ are coordinates.
Signs by Quadrant:
- I: $\sin>0, \cos>0$.
- II: $\sin>0, \cos<0$.
- III: $\sin<0, \cos<0$.
- IV: $\sin<0, \cos>0$.
Right Triangle Definition: $\sin A = \text{opposite}/\text{hypotenuse}$, $\cos A = \text{adjacent}/\text{hypotenuse}$.
Periodicity: $\sin(x+2\pi) = \sin x$, $\cos(x+2\pi) = \cos x$.
Basic Identity: $\sin^2 x + \cos^2 x = 1$.
Co-function Identities: $\cos x = \sin(x+\pi/2)$, $\sin x = \cos(x-\pi/2)$.
Even/Odd Functions: $\sin(-x) = -\sin x$ (odd), $\cos(-x) = \cos x$ (even).
Polar Coordinates: $(x,y) = (r \cos \theta, r \sin \theta)$. $(r, \theta)$ are polar coordinates.

Graph of $\sin x$: Wave-like, oscillates between -1 and 1, period $2\pi$.
Graph of $\cos x$: Wave-like, oscillates between -1 and 1, period $2\pi$. Similar to $\sin x$ shifted by $\pi/2$.

Definition: $\tan x = \sin x / \cos x$. Defined when $\cos x \ne 0$ (i.e., $x \ne \pi/2 + n\pi$).
Right Triangle Definition: $\tan A = \text{opposite}/\text{adjacent}$.
Graph of $\tan x$: Vertical asymptotes at $\pi/2 + n\pi$, period $\pi$.
Applications: Measuring heights/distances.
Other Functions: $\cot x = 1/\tan x$, $\sec x = 1/\cos x$, $\csc x = 1/\sin x$.
Identities: $1 + \tan^2 x = \sec^2 x$.

Theorem 3:
- $\sin(A+B) = \sin A \cos B + \cos A \sin B$.
- $\cos(A+B) = \cos A \cos B - \sin A \sin B$.
Corollary:
- $\sin(A-B) = \sin A \cos B - \cos A \sin B$.
- $\cos(A-B) = \cos A \cos B + \sin A \sin B$.
Double Angle Formulas:
- $\sin(2x) = 2 \sin x \cos x$.
- $\cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$.
Half Angle Formulas:
- $\cos^2 x = (1+\cos 2x)/2$.
- $\sin^2 x = (1-\cos 2x)/2$.

Rotation Matrix: For a point $P=(x,y)$ rotated by angle $\phi$ to $P'=(x',y')$: $$x' = (\cos \phi)x - (\sin \phi)y$$ $$y' = (\sin \phi)x + (\cos \phi)y$$
This can be written as a matrix multiplication: $$\begin{pmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x' \\ y' \end{pmatrix}$$