Genesis's Notes

Deriving the Sum and Difference Formulas Trigonometry · Proof Deriving the Sum & Difference Formulas Every formula in the family flows from a single geometric argument — the distance formula on the unit circle. 📐 4 Steps · 🔑 Distance Formula + Rotation · ⏱ 8 min read 📌 Strategy We derive $\cos(\alpha - \beta)$ first — then everything else follows from it : $\cos(\alpha - \beta)$ → $\cos(\alpha + \beta)$ → $\sin(\alpha \pm \beta)$ → $\tan(\alpha \pm \beta)$ Step 01 Derive $\cos(\alpha - \beta)$ Setup Place two angles $\al...

2.1 Trigonometry Foundations — Complete Knowledge Points Chapter 2 · 2.1  |  Trigonometry Foundations Chapter 2 · Trigonometry 2.1 Trigonometry — Foundations § 5 Topics · Angles · Unit Circle · 6 Trig Functions · ASTC · Special Angles Contents Angle Measurement The Unit Circle Six Trigonometric Functions Signs by Quadrant — ASTC Rule Special Angle Values 01 Angle Measurement There are two ways to measure angles in mathematics. Degrees The system used in high school geometry. A full circle = 360° . Radians The system used in advanced mathematics. A full circle = ...

1.3 Exponential & Logarithmic Functions — Complete Knowledge Points Algebra · 1.3  |  Exponential & Logarithmic Functions Chapter 1 · Algebra 1.3 Exponential & Logarithmic Functions § 7 Topics · Exponents · Exponential · Logarithm · Log Rules · Solving · Natural Log · e Contents Properties of Exponents Exponential Functions Definition of Logarithm Logarithm Rules Solving Exponential Equations Solving Logarithmic Equations Natural Logarithm $\ln x$ and $e$ 01 Properties of Exponents These rules are the foundation of everything in this chapter: Rule Formula Product $a^m \cdot a^n = a^{m+n}$ Quotient $\dfrac{a^m}{a^n} = a^{m-n}$ Power $(a^m)^n = a^{mn}$ Negative $a^{-n} = \dfrac{1}{a^n}$ Fractional ...

1.2 Functions — Complete Knowledge Points Algebra · 1.2  |  Functions — Complete Knowledge Points Chapter 1 · Algebra 1.2 Functions — Complete Knowledge Points § 6 Topics · Domain · Composition · Inverse · Even & Odd · Common Functions Contents What is a Function? Domain and Range Function Composition Inverse Functions Even and Odd Functions Common Functions 01 What is a Function? A function is a rule that assigns exactly one output to every input . Think of it like a machine: ┌──────────┐ x ──▶ │ f(x) │ ──▶ output └──────────┘ The input is called the argument , the output is called the value , and we write it as $f(x)$, read as "f of x" . ⚠️ Key rule: One input must never produce two different out...