Algebra Basics

Algebra

Definition

Algebra is a branch of mathematics that deals with the study of variables and their relationships, often expressed through the use of symbols, equations, and functions.

Key Concepts

• Variables and Constants: Variables are letters or symbols that represent unknown values, while constants are numbers.
• Expressions and Equations: An algebraic expression is a combination of variables, constants, and mathematical operations, while an equation is a statement that two algebraic expressions are equal.
• Linear Equations: Equations in which the highest power of the variable(s) is 1, e.g., 2x + 3 = 5.
• Quadratic Equations: Equations in which the highest power of the variable(s) is 2, e.g., x^2 + 4x + 4 = 0.
• Functions: A relation between a set of inputs (domain) and a set of possible outputs (range).

Algebraic Operations

• Addition and Subtraction: Combining like terms in an algebraic expression.
• Multiplication: Distributive property: a(b + c) = ab + ac.
• Factoring: Expressing an algebraic expression as a product of simpler expressions.

Trigonometry

Definition

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly triangles with right angles (90-degree angles).

Key Concepts

• Angles and Triangles: Measurement of angles in degrees, minutes, and seconds; types of angles (acute, obtuse, right, straight).
• Trigonometric Ratios: Relationships between the sides and angles of a right triangle, including sine, cosine, and tangent (SOH-CAH-TOA).
• Identities and Formulas: Pythagorean identity, sum and difference formulas, and double-angle formulas.

Trigonometric Functions

• Sine (sin): Opposite side over hypotenuse.
• Cosine (cos): Adjacent side over hypotenuse.
• Tangent (tan): Opposite side over adjacent side.
• Cotangent (cot), Secant (sec), and Cosecant (csc): Reciprocal of tangent, cosine, and sine, respectively.

Algebra

• Algebra is a branch of mathematics that deals with the study of variables and their relationships.
• Variables are letters or symbols that represent unknown values, while constants are numbers.
• Algebraic expressions are combinations of variables, constants, and mathematical operations.
• Equations are statements that two algebraic expressions are equal.

Equations

• Linear equations have the highest power of the variable(s) as 1, e.g., 2x + 3 = 5.
• Quadratic equations have the highest power of the variable(s) as 2, e.g., x^2 + 4x + 4 = 0.

Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range).

Algebraic Operations

• Addition and Subtraction: Combining like terms in an algebraic expression.
• Multiplication: Distributive property: a(b + c) = ab + ac.
• Factoring: Expressing an algebraic expression as a product of simpler expressions.

Trigonometry

• Trigonometry deals with the relationships between the sides and angles of triangles, particularly triangles with right angles (90-degree angles).

Angles and Triangles

• Angles are measured in degrees, minutes, and seconds.
• Types of angles include acute, obtuse, right, and straight angles.

Trigonometric Ratios

• Sine (sin): Opposite side over hypotenuse.
• Cosine (cos): Adjacent side over hypotenuse.
• Tangent (tan): Opposite side over adjacent side.

Trigonometric Identities and Formulas

• Pythagorean identity.
• Sum and difference formulas.
• Double-angle formulas.

Additional Trigonometric Functions

• Cotangent (cot): Reciprocal of tangent.
• Secant (sec): Reciprocal of cosine.
• Cosecant (csc): Reciprocal of sine.