Introduction to Algebra

• Math encompasses a broad field studying quantity, structure, space, and change.
• It includes arithmetic, algebra, geometry, trigonometry, calculus, and statistics.

Algebra

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities.
• It involves solving equations and inequalities to find unknown values.
• Variables are symbols, often letters, that represent unknown quantities.
• Constants are fixed values that do not change.
• Expressions are combinations of variables, constants, and operations like addition, subtraction, multiplication, and division
• Equations are mathematical statements that show the equality between two expressions.
• Formulas are equations that express a general rule or relationship.
• Linear equations can be written in the form ax + b = 0, where a and b are constants and x is the variable.
• Quadratic equations can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.
• Systems of equations involve two or more equations with the same variables.
• The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously.
• Factoring is the process of breaking down an expression into a product of simpler expressions.
• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• The degree of a polynomial is the highest power of the variable in the polynomial.
• Exponents indicate the number of times a base is multiplied by itself.
• Radicals, such as square roots and cube roots, are used to find a number that, when raised to a power, equals a given value.
• Logarithms are the inverse of exponential functions.
• Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit defined as the square root of -1.
• Matrices are rectangular arrays of numbers arranged in rows and columns, used to solve systems of equations and perform linear transformations.

Geometry

• Geometry is the study of shapes, sizes, and positions of figures in space.
• Euclidean geometry deals with flat planes and figures with straight lines and curves.
• Points are locations in space, having no dimension.
• Lines are straight paths extending infinitely in both directions.
• Planes are flat surfaces extending infinitely in all directions.
• Line segments are parts of a line with two endpoints.
• Rays are parts of a line with one endpoint, extending infinitely in one direction.
• Angles are formed by two rays sharing a common endpoint (vertex), measured in degrees or radians.
• Acute angles are less than 90 degrees.
• Right angles are exactly 90 degrees.
• Obtuse angles are greater than 90 degrees but less than 180 degrees.
• Straight angles are exactly 180 degrees.
• Complementary angles add up to 90 degrees.
• Supplementary angles add up to 180 degrees.
• Triangles are polygons with three sides and three angles.
• Equilateral triangles have three equal sides and three equal angles (60 degrees each).
• Isosceles triangles have two equal sides and two equal angles.
• Scalene triangles have no equal sides and no equal angles.
• Right triangles have one right angle.
• The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (a^2 + b^2 = c^2).
• Quadrilaterals are polygons with four sides and four angles.
• Squares have four equal sides and four right angles.
• Rectangles have opposite sides equal and four right angles.
• Parallelograms have opposite sides parallel and equal.
• Rhombuses have four equal sides and opposite angles equal.
• Trapezoids have at least one pair of parallel sides.
• Circles are sets of points equidistant from a center point.
• The radius is the distance from the center to any point on the circle.
• The diameter is the distance across the circle through the center (twice the radius).
• The circumference is the distance around the circle (2πr or πd).
• The area of a circle is πr^2.
• Polygons are closed figures formed by line segments.
• Regular polygons have all sides and all angles equal.
• Congruent figures have the same shape and size.
• Similar figures have the same shape but different sizes.
• Transformations include translations (sliding), rotations (turning), reflections (flipping), and dilations (scaling).
• Area is the measure of the surface enclosed by a two-dimensional figure.
• Volume is the measure of the space occupied by a three-dimensional object.
• Surface area is the total area of the surfaces of a three-dimensional object.

Trigonometry

• Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles.
• It is particularly useful for solving problems involving right triangles.
• Trigonometric functions relate angles to the ratios of the sides of a right triangle.
• Sine (sin) is the ratio of the length of the opposite side to the length of the hypotenuse.
• Cosine (cos) is the ratio of the length of the adjacent side to the length of the hypotenuse.
• Tangent (tan) is the ratio of the length of the opposite side to the length of the adjacent side.
• Cosecant (csc) is the reciprocal of sine (hypotenuse/opposite).
• Secant (sec) is the reciprocal of cosine (hypotenuse/adjacent).
• Cotangent (cot) is the reciprocal of tangent (adjacent/opposite).
• These functions are often remembered using the acronym SOH CAH TOA (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent).
• The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane, used to define trigonometric functions for all angles.
• Angles can be measured in degrees or radians.
• Radians are a unit of angular measure where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.
• Trigonometric identities are equations that are true for all values of the variables for which the expressions are defined.
• The Pythagorean identity is sin^2(θ) + cos^2(θ) = 1.
• The law of sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant (a/sin(A) = b/sin(B) = c/sin(C)).
• The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles (c^2 = a^2 + b^2 - 2ab cos(C)).
• Trigonometric equations are equations that involve trigonometric functions and need to be solved for the unknown angle.
• Inverse trigonometric functions (arcsin, arccos, arctan) find the angle that corresponds to a given trigonometric ratio.
• Trigonometry is used in navigation, surveying, engineering, and physics to solve problems involving angles and distances.