Introduction to Algebra

• Math encompasses a broad range of topics including algebra, geometry, and trigonometry

Algebra

• Focuses on symbols and the rules for manipulating these symbols
• These symbols represent numbers whose values are either unknown or can change
• Key concepts include variables, expressions, equations, and inequalities
• Variables are symbols (usually letters) that represent unknown or changeable values
• Expressions are combinations of variables, numbers, and operations
• Equations are statements that two expressions are equal
• Inequalities compare two expressions using symbols like <, >, ≤, or ≥
• Solving equations and inequalities involves finding the values of the variables that make the statement true
• Linear equations can be written in the form ax + b = c, where x is the variable and a, b, and c are constants
• Quadratic equations can be written in the form ax² + bx + c = 0, where x is the variable and a, b, and c are constants
• Systems of equations involve two or more equations with the same variables
• Algebraic manipulation involves rearranging expressions and equations using rules of arithmetic and algebra to solve for unknown variables or simplify expressions
• Factoring is the process of breaking down an expression into a product of simpler expressions
• Expanding is the process of multiplying out terms in an expression

Geometry

• Deals with the properties and relationships of points, lines, surfaces, and solids
• Key concepts include shapes, sizes, relative positions of figures, and the properties of space
• Euclidean geometry is based on a set of axioms and postulates established by the Greek mathematician Euclid
• Points are exact locations in space and have no dimension
• Lines are straight paths that extend infinitely in both directions and have one dimension
• Planes are flat surfaces that extend infinitely in all directions and have two dimensions
• Angles are formed by two rays sharing a common endpoint (vertex)
• Triangles are polygons with three sides and three angles.
• Common types include equilateral, isosceles, and right triangles
• Quadrilaterals are polygons with four sides and four angles
• Common types include squares, rectangles, parallelograms, and trapezoids
• Circles are sets of points equidistant from a center point
• Key elements include radius, diameter, circumference, and area
• Area is the measure of the surface enclosed by a two-dimensional figure
• Perimeter is the total distance around the boundary of 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
• 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: a² + b² = c²
• Geometric transformations include translations (sliding), rotations (turning), reflections (flipping), and dilations (scaling)
• Congruence means that two figures have the same shape and size
• Similarity means that two figures have the same shape but different sizes

Trigonometry

• Studies the relationships between the angles and sides of triangles
• Focuses on trigonometric functions such as sine, cosine, and tangent
• Trigonometric functions relate the angles of a right triangle to the ratios of its sides
• 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
• The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane
• Trigonometric functions can be defined for angles greater than 90 degrees using the unit circle
• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables
• Examples include sin²(x) + cos²(x) = 1 and tan(x) = sin(x) / cos(x)
• Sine rule: a/sin(A) = b/sin(B) = c/sin(C), which relates the sides and angles of any triangle
• Cosine rule: c² = a² + b² - 2ab cos(C), which relates the sides and angles of any triangle
• Trigonometric equations involve finding the values of the angles that satisfy the equation
• Inverse trigonometric functions (arcsin, arccos, arctan) find the angle that corresponds to a given trigonometric ratio
• Trigonometry is used in various fields such as surveying, navigation, and physics