Mathematical Facts, Constants & Formulas
Grade 11: Complex Numbers, Logarithms, Unit Circle & Series
~11 min read
HESTEN ACADEMIC REFERENCE ARCHIVE
CALL NO: QA154.H47 2026 • DEWEY: 512.9042 • VOL. XI
Volume XI • Advanced Algebra & Trigonometry Codex
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
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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.
§ 11.1
The Field of Complex Numbers ($\mathbb{C}$)
DEFINITION 11.1.1
Topic: The Imaginary Unit & Argand Plane
The imaginary unit $i$ satisfies $i = \sqrt{-1} \implies i^2 = -1$.
$$\text{Complex Number: } z = a + bi \quad (a, b \in \mathbb{R}, \ a = \text{Re}(z), \ b = \text{Im}(z))$$
$$\text{Complex Conjugate: } \bar{z} = a - bi \implies z \cdot \bar{z} = a^2 + b^2 \in \mathbb{R}^+$$
$$\text{Modulus (Magnitude): } |z| = \sqrt{a^2 + b^2}$$
Powers of $i$ Modulo 4
$$i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1$$
Divide power $k$ by $4$: remainder determines power ($i^{27} \implies 27 \equiv 3 \pmod 4 \implies -i$).
Complex Division via Conjugate
$$\frac{z_1}{z_2} = \frac{a + bi}{c + di} \times \frac{c - di}{c - di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$$
Multiply numerator and denominator by denominator's conjugate to clear $i$ from denominator.
§ 11.2
Polynomial Theorems & Synthetic Division
THEOREM 11.2.1
Topic: The Remainder & Rational Root Theorems
- 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).
§ 11.3
Rational Functions & Asymptotes
| 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 |
§ 11.4
Logarithm Laws & Natural Logarithms ($\ln x$)
Product to Sum
$$\log_b(xy) = \log_b x + \log_b y$$
Multiplication inside turns into addition outside.
Quotient to Difference
$$\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$$
Division inside turns into subtraction outside.
Exponent Jumping
$$\log_b(x^k) = k \cdot \log_b x$$
Brings down variables stuck in exponents!
Base Conversion
$$\log_b x = \frac{\ln x}{\ln b} = \frac{\log_{10} x}{\log_{10} b}$$
Computes logs of arbitrary bases on standard calculators.
§ 11.5
The Complete Unit Circle ($x = \cos\theta, \ y = \sin\theta$)
| 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 |
§ 11.6
Advanced Trigonometric Identities
Pythagorean Identities
$$\begin{aligned} \sin^2\theta + \cos^2\theta &= 1 \\ 1 + \tan^2\theta &= \sec^2\theta \\ 1 + \cot^2\theta &= \csc^2\theta \end{aligned}$$
Fundamental relations derived from unit circle circle equation $x^2 + y^2 = 1$.
Double-Angle Formulas
$$\begin{aligned} \sin(2\theta) &= 2\sin\theta\cos\theta \\ \cos(2\theta) &= \cos^2\theta - \sin^2\theta \\ &= 2\cos^2\theta - 1 = 1 - 2\sin^2\theta \end{aligned}$$
Expands multiple-angle arguments into single-angle factors.
§ 11.7
Periodic Function Graphing: Sinusoidal Waves
General Wave Equation
$$y = A \sin(Bx - C) + D$$
Amplitude: $|A|$ •
Period ($T$): $\frac{2\pi}{B}$ •
Phase Shift: $\frac{C}{B}$ •
Midline: $y = D$.
§ 11.8
Sequences & Infinite Geometric Series ($\Sigma$)
Finite Arithmetic Series
$$S_n = \frac{n(a_1 + a_n)}{2} = \frac{n}{2}[2a_1 + (n-1)d]$$
Sum of first $n$ terms with common difference $d$.
Finite Geometric Series
$$S_n = \frac{a_1(1 - r^n)}{1 - r} \quad (r \neq 1)$$
Sum of first $n$ terms with common ratio $r$.
Infinite Geometric Series Sum
$$S_\infty = \sum_{k=1}^\infty a_1 r^{k-1} = \frac{a_1}{1 - r} \iff |r| < 1$$
If $|r| \ge 1$, the infinite series diverges strictly to $\pm\infty$.
§ 11.9
Combinatorics, Binomial Expansion & Normal Distribution
Combinations ($nCr$)
$$\binom{n}{r} = \frac{n!}{r!(n - r)!}$$
Pascal's Triangle coefficients for binomial expansion.
Binomial Expansion Formula
$$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$
Expands $(a+b)^n$ into $n+1$ distinct terms.
68-95-99.7 Rule & $z$-Scores
$$z = \frac{x - \mu}{\sigma}$$
$68.2\%$ within $1\sigma$, $95.4\%$ within $2\sigma$, $99.7\%$ within $3\sigma$.
§ 11.10
Scholia & Computational Mental Math Speed Hacks
HACK I
"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!
HACK II
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.
HACK III
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.
HACK IV
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!
§ 11.11
The Grand Synoptic Tables & Complete Grade 11 / Algebra II Student Cheat Sheet
Grade 11 / Algebra II Master Reference Concordance
Authorized curriculum reference concordance • Algebra II & Trigonometry Complete
Complex Numbers
- $i = \sqrt{-1}, \ i^2 = -1, \ i^3 = -i, \ i^4 = 1$
- $z = a + bi \quad \vert \quad \bar{z} = a - bi$
- $|z| = \sqrt{a^2 + b^2}$
- $z \cdot \bar{z} = a^2 + b^2$
- Divide: Multiply by $\frac{\bar{z}}{\bar{z}}$
Logarithm Laws
- $\log_b(xy) = \log_b x + \log_b y$
- $\log_b(x/y) = \log_b x - \log_b y$
- $\log_b(x^k) = k\log_b x$
- Change of base: $\frac{\ln x}{\ln b}$
- $\ln(e) = 1, \ \ln(1) = 0, \ e^{\ln x} = x$
Unit Circle & ASTC
- $(x, y) = (\cos\theta, \sin\theta)$ on $r=1$
- ASTC: Q1 All, Q2 Sin, Q3 Tan, Q4 Cos
- $\pi \text{ rad} = 180^\circ$
- $30^\circ = \frac{\pi}{6}, 45^\circ = \frac{\pi}{4}, 60^\circ = \frac{\pi}{3}$
- $\tan\theta = \frac{\sin\theta}{\cos\theta}$
Trig Identities
- $\sin^2\theta + \cos^2\theta = 1$
- $1 + \tan^2\theta = \sec^2\theta$
- $\sin(2\theta) = 2\sin\theta\cos\theta$
- $\cos(2\theta) = \cos^2\theta - \sin^2\theta$
- Period $T = \frac{2\pi}{B}$
Sequences & Series
- 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: $P(c) = 0 \iff (x-c) \text{ factor}$
Combinatorics & Stats
- 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: $\text{deg}(P) = \text{deg}(Q) \implies y = \frac{a_n}{b_n}$