$$z=a+bi,\quad i^2=-1,\quad\text{Re}(z)=a,\quad\text{Im}(z)=b.$$$$|z|=\sqrt{a^2+b^2},\quad\arg(z)=\arctan(b/a)\text{ (adjusted for quadrant)}.$$$$\bar{z}=a-bi,\quad z\bar{z}=|z|^2,\quad\frac{1}{z}=\frac{\bar{z}}{|z|^2}.$$

$$\frac{p(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b},\quad\frac{p(x)}{(x-a)(x^2+b)}=\frac{A}{x-a}+\frac{Bx+C}{x^2+b}.$$

A system of up to three linear equations in three unknowns can be solved by elimination: combine pairs of equations to cancel one variable at a time, reducing to a smaller system, until a single equation in one unknown remains — then substitute back up.

Geometrically, each equation $ax+by+cz=d$ is a plane in 3D (Chapter 3 of this track studies these directly). Solving the system asks how three planes intersect:

• Unique solution — the three planes meet at exactly one point (the generic case).
• No solution — elimination produces a contradiction like $0=k$ for $k\neq0$; geometrically, no point lies on all three planes at once (e.g. two are parallel, or the three form a triangular "tube" with no common point).
• Infinitely many solutions — elimination produces an identity like $0=0$, leaving a free parameter; geometrically, the planes share a common line (or are all the same plane).

$$|f(x)|:\text{ reflect negative part above }x\text{-axis}.$$$$f(|x|):\text{ reflect positive part across }y\text{-axis}.$$

Equations $|f(x)|=k$ give two cases: $f(x)=k$ and $f(x)=-k.$ Inequalities $|f(x)|<k$ give $-k<f(x)<k.$

The compound angle formulae extend SL's double-angle identities (Chapter 3 of the SL track) to a sum or difference of any two angles: $$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B,\qquad\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B,$$ $$\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}.$$ Setting $A=B=\theta$ in the first two recovers exactly the SL double-angle identities $\sin2\theta=2\sin\theta\cos\theta$ and $\cos2\theta=\cos^2\theta-\sin^2\theta$ — those are this chapter's special case, not a separate fact.

The reciprocal trigonometric functions are $$\sec\theta=\frac{1}{\cos\theta},\qquad\csc\theta=\frac{1}{\sin\theta},\qquad\cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}.$$ Dividing the Pythagorean identity $\sin^2\theta+\cos^2\theta=1$ through by $\cos^2\theta$ (then by $\sin^2\theta$) gives two more standard identities: $$1+\tan^2\theta=\sec^2\theta,\qquad1+\cot^2\theta=\csc^2\theta.$$

$$\mathbf{v}=\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix},\quad|\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2},\quad\mathbf{u}\cdot\mathbf{v}=u_1v_1+u_2v_2+u_3v_3.$$

Line: $\mathbf{r}=\mathbf{a}+t\mathbf{d}.$ Parametric: $x=a_1+td_1,\;y=a_2+td_2,\;z=a_3+td_3.$

Plane: $\mathbf{r}\cdot\hat{\mathbf{n}}=d,$ or $ax+by+cz=d,$ where $\mathbf{n}=\langle a,b,c\rangle$ is the normal vector.

Angle between planes: $\cos\theta=\dfrac{|\mathbf{n}_1\cdot\mathbf{n}_2|}{|\mathbf{n}_1||\mathbf{n}_2|}.$

Distance from point $P$ to plane $ax+by+cz=d$: $\dfrac{|aP_x+bP_y+cP_z-d|}{\sqrt{a^2+b^2+c^2}}.$

$$\hat{\mu}=\bar{x}=\frac{\sum x_i}{n}\quad(\text{unbiased for }\mu).$$$$S^2=\frac{\sum(x_i-\bar{x})^2}{n-1}\quad(\text{unbiased for }\sigma^2;\;\text{use }s_{n-1}\text{ on GDC}).$$

$$X\sim\text{Po}(\lambda):\quad P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!},\quad\mu=\sigma^2=\lambda.$$

Conditions: events occur independently at a constant average rate $\lambda$ per unit interval.

Poisson as approximation to Binomial: when $n$ is large, $p$ is small, and $np=\lambda$ is moderate.

A function $f$ is continuous at $x = c$ if and only if all three conditions of the Continuity Checklist are satisfied:

1. Condition 1: $f(c)$ is defined (meaning $c$ is in the domain of $f$).
2. Condition 2: $\lim_{x\to c} f(x)$ exists (meaning left and right limits are finite and agree: $\lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x) = L$).
3. Condition 3: $\lim_{x\to c} f(x) = f(c)$ (the limiting trend matches the actual output).

Failure of any condition produces a specific discontinuity geometry on the Cartesian plane:

• Case 1 · Continuous: All 3 conditions hold. The curve passes unbroken through $(c, f(c))$.
• Case 2 · Removable (Hole): Condition 1 fails (undefined at $c$), but Condition 2 holds (limit $L$ exists). The graph has an open ring at $(c, L)$.
• Case 3 · Removable (Displaced Point): Conditions 1 and 2 hold, but Condition 3 fails ($f(c) \neq L$). An open ring sits at $(c, L)$ while a solid dot sits at $(c, f(c))$.
• Case 4 · Jump Discontinuity: Condition 2 fails because one-sided limits are finite but unequal ($\lim_{x\to c^-} f(x) \neq \lim_{x\to c^+} f(x)$). A step break occurs in the graph.
• Case 5 · Infinite Discontinuity (Vertical Asymptote): Condition 2 fails because one or both one-sided limits grow without bound to $\pm\infty$.
• Case 6 · Essential / Oscillating Discontinuity: Condition 2 fails because function values oscillate infinitely rapidly between bounds as $x$ approaches $c$.