Eleventh Grade Mathematics Reference Codex (Algebra II & Trigonometry)
The Complete Scholastic Guide to Complex Numbers ($\mathbb{C}$), Polynomial Division, Logarithmic Laws, The Full Unit Circle, Analytic Trigonometry, Infinite Series & Synoptic Concordance
❦
An exhaustive collegiate-preparatory reference compendium designed for eleventh-grade scholars, advanced mathematics educators, and STEM practitioners. Codifying field operations over the complex numbers, polynomial ring factorization, transcendental logarithmic functions, circular periodic metrics, convergent geometric series, and computational algebra heuristics.
The imaginary unit $i$ satisfies $i = \sqrt{-1} \implies i^2 = -1$.
$$\text{Complex Number: } z = a + bi \quad (a, b \in \mathbb{R}, \ a = \text{Re}(z), \ b = \text{Im}(z))$$
$$\text{Complex Conjugate: } \bar{z} = a - bi \implies z \cdot \bar{z} = a^2 + b^2 \in \mathbb{R}^+$$
$$\text{Modulus (Magnitude): } |z| = \sqrt{a^2 + b^2}$$
Polynomial Remainder Theorem: Dividing $P(x)$ by $(x - c)$ yields remainder $R = P(c)$.Factor Theorem: Binomial $(x - c)$ is a factor of $P(x) \iff P(c) = 0$.Rational Root Theorem: Any rational root of $a_n x^n + \dots + a_0 = 0$ must have form $\pm \frac{p}{q}$, where $p \mid a_0$ and $q \mid a_n$.Fundamental Theorem of Algebra: Every degree-$n$ polynomial has exactly $n$ roots in $\mathbb{C}$ (counting multiplicity).
Asymptote Type
Condition on $f(x) = \frac{P(x)}{Q(x)}$
Behavior & Equation
Vertical Asymptote (VA) $Q(c) = 0$ and $P(c) \neq 0$ Line $x = c$ (Infinite discontinuity) Removable Hole $P(c) = 0$ and $Q(c) = 0$ (Common factor cancels) Point hole at $(c, \lim_{x\to c} f(x))$ Horizontal (Degree Top < Bottom) $\text{deg}(P) < \text{deg}(Q)$ Line $y = 0$ ($x$-axis) Horizontal (Equal Degrees) $\text{deg}(P) = \text{deg}(Q)$ Line $y = \frac{a_{\text{lead}}}{b_{\text{lead}}}$ Slant / Oblique Asymptote $\text{deg}(P) = \text{deg}(Q) + 1$ Line $y = mx + b$ obtained via polynomial division quotient
Degrees
Radians
Coordinate $(x, y) = (\cos\theta, \sin\theta)$
$\tan\theta = y/x$
Quadrant
$0^\circ$ $0$ $(1, 0)$ $0$ Axis $30^\circ$ $\frac{\pi}{6}$ $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$ $\frac{\sqrt{3}}{3}$ QI $45^\circ$ $\frac{\pi}{4}$ $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$ $1$ QI $60^\circ$ $\frac{\pi}{3}$ $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ $\sqrt{3}$ QI $90^\circ$ $\frac{\pi}{2}$ $(0, 1)$ $\text{Undefined}$ Axis $120^\circ$ $\frac{2\pi}{3}$ $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ $-\sqrt{3}$ QII $135^\circ$ $\frac{3\pi}{4}$ $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$ $-1$ QII $150^\circ$ $\frac{5\pi}{6}$ $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$ $-\frac{\sqrt{3}}{3}$ QII $180^\circ$ $\pi$ $(-1, 0)$ $0$ Axis $210^\circ$ $\frac{7\pi}{6}$ $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$ $\frac{\sqrt{3}}{3}$ QIII $225^\circ$ $\frac{5\pi}{4}$ $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$ $1$ QIII $240^\circ$ $\frac{4\pi}{3}$ $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$ $\sqrt{3}$ QIII $270^\circ$ $\frac{3\pi}{2}$ $(0, -1)$ $\text{Undefined}$ Axis $300^\circ$ $\frac{5\pi}{3}$ $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$ $-\sqrt{3}$ QIV $315^\circ$ $\frac{7\pi}{4}$ $\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$ $-1$ QIV $330^\circ$ $\frac{11\pi}{6}$ $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$ $-\frac{\sqrt{3}}{3}$ QIV $360^\circ$ $2\pi$ $(1, 0)$ $0$ Axis
"All Students Take Calculus" (ASTC)
$$\mathbf{A} \text{ (All +)} \quad \vert \quad \mathbf{S} \text{ (Sin +)} \quad \vert \quad \mathbf{T} \text{ (Tan +)} \quad \vert \quad \mathbf{C} \text{ (Cos +)}$$
QI: All positive • QII: Sine only • QIII: Tangent only • QIV: Cosine only. Instant sign check!
Synthetic Division 10-Second Speed Run
$$\text{Divisor } (x - c) \implies \text{Run synthetic with root } c$$
Never use bulky long division for linear divisors $(x-c)$. Synthetic division yields $P(c)$ in 3 rapid rows.
Log Power Dropping Shortcut
$$b^x = C \implies x = \frac{\ln C}{\ln b}$$
Take natural log of both sides to instantly pop unknown powers down to the base line.
Infinite Series Convergence Sanity Test
$$\text{If } |r| \ge 1 \implies \text{Diverges immediately! Do not apply } \frac{a_1}{1-r}$$
Formula $\frac{a_1}{1-r}$ is valid ONLY when $|r| < 1$. Check ratio first!
$i = \sqrt{-1}, \ i^2 = -1, \ i^3 = -i, \ i^4 = 1$
$z = a + bi \quad \vert \quad \bar{z} = a - bi$
$|z| = \sqrt{a^2 + b^2}$
$z \cdot \bar{z} = a^2 + b^2$
Divide: Multiply by $\frac{\bar{z}}{\bar{z}}$
$\log_b(xy) = \log_b x + \log_b y$
$\log_b(x/y) = \log_b x - \log_b y$
$\log_b(x^k) = k\log_b x$
Change of base: $\frac{\ln x}{\ln b}$
$\ln(e) = 1, \ \ln(1) = 0, \ e^{\ln x} = x$
$(x, y) = (\cos\theta, \sin\theta)$ on $r=1$
ASTC: Q1 All, Q2 Sin, Q3 Tan, Q4 Cos
$\pi \text{ rad} = 180^\circ$
$30^\circ = \frac{\pi}{6}, 45^\circ = \frac{\pi}{4}, 60^\circ = \frac{\pi}{3}$
$\tan\theta = \frac{\sin\theta}{\cos\theta}$
$\sin^2\theta + \cos^2\theta = 1$
$1 + \tan^2\theta = \sec^2\theta$
$\sin(2\theta) = 2\sin\theta\cos\theta$
$\cos(2\theta) = \cos^2\theta - \sin^2\theta$
Period $T = \frac{2\pi}{B}$
Arithmetic: $S_n = \frac{n(a_1+a_n)}{2}$
Geometric: $S_n = \frac{a_1(1-r^n)}{1-r}$
Infinite Sum: $S_\infty = \frac{a_1}{1-r} \ (|r| < 1)$
Remainder Thm: $P(c) = R$
Factor Thm: $P(c) = 0 \iff (x-c) \text{ factor}$
Combinations: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$
$(a+b)^n = \sum \binom{n}{k}a^{n-k}b^k$
$z$-score: $z = \frac{x - \mu}{\sigma}$
Empirical: $68\% - 95\% - 99.7\%$
Asymptote: $\text{deg}(P) = \text{deg}(Q) \implies y = \frac{a_n}{b_n}$