Geometry Introduction
Trigonometric Ratios, Functions, Coordinates, and Vectors (High School Level)
Overview
This introduction covers the fundamentals of geometry studied in high school mathematics. Starting from trigonometric ratios and functions, we learn to represent figures on the coordinate plane and study the concept of vectors. These form the foundation for geometry at the university level and beyond.
Learning Objectives
- Understand the definitions and basic formulas of trigonometric ratios and functions
- Express lines and circles using equations on the coordinate plane
- Understand vector operations and their geometric meaning
- Calculate angles and lengths using the dot product
Table of Contents
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Chapter 1 Trigonometric Ratios
Definitions of sin, cos, tan; reciprocal relations; law of sines and cosines
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Chapter 2
Trigonometric Functions
General angles, radians, graphs and properties of trigonometric functions
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Chapter 3 Fundamental Trigonometric Identity
Proof of $\sin^2\theta + \cos^2\theta = 1$
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Chapter 4 Addition Formulas
Proofs of $\sin(\alpha+\beta)$ and $\cos(\alpha+\beta)$
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Chapter 5
Double-Angle and Half-Angle Formulas
Derivation of double-angle, half-angle, and triple-angle formulas
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Chapter 6 Product-to-Sum and Sum-to-Product Formulas
Deriving product-to-sum and sum-to-product formulas from addition formulas
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Chapter 7 Complementary Angle Formulas
Relations between sin, cos, tan and their cofunctions
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Chapter 8 Coordinate Plane and Lines
Points and lines, distance, equations of lines
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Chapter 9 Equations of Circles
Standard form, tangent lines, positional relationships between circles and lines
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Chapter 10 Vectors
Definition, operations, component representation, dot product
Topic Articles
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Heron's Formula
Computing the area of a triangle from its three sides: algebraic proof and visual proof using the incircle
Prerequisites
- Middle school geometry (triangles, circles, similarity)
- Quadratic equations and functions
- Square root calculations