Algebra Overview

Definition: A branch of mathematics dealing with symbols and the rules for manipulating those symbols.
Key Concepts:
- Variables: Symbols (often x, y, z) used to represent numbers.
- Expressions: Combinations of variables, numbers, and operations (e.g., 2x + 3).
- Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
- Functions: Relationships between inputs and outputs, often written as f(x).
Operations:
- Addition and Subtraction: Combine like terms to simplify expressions.
- Multiplication and Division: Apply distributions and factorization.
- Solving Equations: Techniques include isolating variables, substitution, and using inverse operations.
Types of Equations:
- Linear Equations: Form y = mx + b; graphs as straight lines.
- Quadratic Equations: Form ax² + bx + c = 0; can be solved using factoring, completing the square, or quadratic formula.
- Polynomial Equations: Involves terms with exponents; can include various degrees.

Definition: A branch of mathematics dealing with relationships between the angles and sides of triangles.
Key Concepts:
- Trigonometric Ratios: Ratios of sides in a right triangle.
  - Sine (sin) = Opposite/Hypotenuse
  - Cosine (cos) = Adjacent/Hypotenuse
  - Tangent (tan) = Opposite/Adjacent
Key Functions:
- Reciprocal Functions:
  - Cosecant (csc) = 1/sin
  - Secant (sec) = 1/cos
  - Cotangent (cot) = 1/tan
Unit Circle:
- Defines trigonometric functions for all angles; crucial for understanding function behavior.
- Coordinates of points (cos(θ), sin(θ)) correspond to angles θ in standard position.
Laws:
- Pythagorean Theorem: a² + b² = c², relates sides of right triangles.
- Law of Sines: (a/sin A) = (b/sin B) = (c/sin C); relates ratios of sides to angles.
- Law of Cosines: c² = a² + b² - 2ab(cos C); generalizes Pythagorean theorem for non-right triangles.
Applications: Used in physics, engineering, computer science, and architecture. Commonly applied in wave functions, oscillations, and circular motion.

Definition: Branch of mathematics focusing on symbols and the rules for manipulating them.
Key Concepts:
- Variables: Symbols (often x, y, z) representing unknown numbers.
- Expressions: Combinations of variables, numbers, and operations (e.g., 2x + 3).
- Equations: Statements declaring two expressions as equal (e.g., 2x + 3 = 7).
- Functions: Relationships between inputs and outputs, often written as f(x).
- Operations:
  - Addition and Subtraction: Simplify expressions by combining like terms.
  - Multiplication and Division: Apply distributions and factorization.
- Solving Equations: Techniques include isolating variables, substitution, and using inverse operations.
- Types of Equations:
  - Linear Equations: Form y = mx + b, with graphs as straight lines.
  - Quadratic Equations: Form ax² + bx + c = 0, solved through factoring, completing the square, or the quadratic formula.
  - Polynomial Equations: Include terms with exponents, categorized by their degrees.

Definition: Branch of mathematics dedicated to exploring relationships between angles and sides of triangles.
Key Concepts:
- Trigonometric Ratios: Ratios of sides in a right triangle.
  - Sine (sin) = Opposite/Hypotenuse.
  - Cosine (cos) = Adjacent/Hypotenuse.
  - Tangent (tan) = Opposite/Adjacent.
Key Functions:
- Reciprocal Functions:
  - Cosecant (csc) = 1/sin.
  - Secant (sec) = 1/cos.
  - Cotangent (cot) = 1/tan.
Unit Circle:
- Defines trigonometric functions for all angles, providing a visual understanding of their behavior.
- Coordinates of points (cos(θ), sin(θ)) represent angles θ in standard position.
Laws:
- Pythagorean Theorem: a² + b² = c², relating the sides of a right triangle.
- Law of Sines: (a/sin A) = (b/sin B) = (c/sin C), connecting ratios of sides to angles.
- Law of Cosines: c² = a² + b² - 2ab(cos C), generalizing the Pythagorean theorem for non-right triangles.
Applications: Used extensively in Physics, Engineering, Computer Science, and Architecture, particularly in understanding wave functions, oscillations, and circular motion.