Trigonometry Chapter 1

Trigonometry

Angles and Triangles

• Measure of angles in degrees, radians, or gradients
• Types of angles: acute, right, obtuse, straight, reflex
• Properties of triangles: sum of angles = 180°, Pythagorean theorem

Trigonometric Ratios

• Sine (sin), cosine (cos), and tangent (tan) of an angle
• Relationships between ratios: sin(A) = cos(90° - A), tan(A) = sin(A) / cos(A)
• Graphs of sine, cosine, and tangent functions

Identities and Formulas

• Pythagorean identity: sin^2(A) + cos^2(A) = 1
• Sum and difference formulas: sin(A+B), cos(A+B), sin(A-B), cos(A-B)
• Double and triple angle formulas

Applications

• Triangulation: solving triangles using trigonometric ratios
• Waves and periodic motion: modeling using sine and cosine functions
• Analytic geometry: using trigonometry to solve problems in 2D and 3D space

Calculus

Limits

• Concept of a limit: approaching a value as the input gets arbitrarily close
• Properties of limits: sum, product, and chain rule
• One-sided and two-sided limits

Derivatives

• Definition of a derivative: rate of change of a function
• Rules of differentiation: power rule, product rule, quotient rule, and chain rule
• Geometric interpretation: tangent lines and slopes

Applications of Derivatives

• Finding maxima and minima: optimization problems
• Motion along a line, curve, and surface: velocity and acceleration
• Related rates: finding rates of change in related quantities

Integrals

• Definition of a definite integral: accumulation of a function over an interval
• Fundamental Theorem of Calculus: relating derivatives and integrals
• Substitution method and integration by parts

Applications of Integrals

• Area under curves: finding areas of regions bounded by curves
• Volume of solids: finding volumes of solids with known cross-sections
• Work and energy: calculating work and energy in physical systems

Algebra

Equations and Inequalities

• Linear equations: solving equations of the form ax + by = c
• Quadratic equations: solving equations of the form ax^2 + bx + c = 0
• Systems of linear equations: solving systems of equations using substitution, elimination, and matrices

Functions

• Domain and range: defining the input and output of a function
• Composition of functions: combining functions using the output of one function as the input of another
• Inverse functions: finding the inverse of a function

Graph Theory

• Basic concepts: vertices, edges, and graphs
• Types of graphs: simple, weighted, directed, and undirected graphs
• Graph algorithms: traversing, searching, and sorting graphs

Linear Algebra

• Vectors: operations, properties, and applications
• Matrices: operations, properties, and applications
• Linear transformations: matrix representations and applications

Trigonometry

Angles and Triangles

• Angles can be measured in degrees, radians, or gradients
• Angles can be classified as acute, right, obtuse, straight, or reflex
• The sum of angles in a triangle is always 180°
• The Pythagorean theorem relates the lengths of sides in a right-angled triangle

Trigonometric Ratios

• Sine (sin), cosine (cos), and tangent (tan) are ratios of triangle sides
• sin(A) = cos(90° - A) and tan(A) = sin(A) / cos(A)
• Sine, cosine, and tangent functions have distinct graphs

Identities and Formulas

• The Pythagorean identity is sin^2(A) + cos^2(A) = 1
• Sum and difference formulas exist for sine and cosine functions
• Double and triple angle formulas are used to simplify trigonometric expressions

Applications

• Trigonometry can be used to solve triangles using triangulation
• Sine and cosine functions model waves and periodic motion
• Trigonometry is used in analytic geometry to solve problems in 2D and 3D space

Calculus

Limits

• A limit is the value approached as the input gets arbitrarily close
• Limit properties include the sum, product, and chain rule
• One-sided and two-sided limits have different properties

Derivatives

• A derivative is the rate of change of a function
• The power rule, product rule, quotient rule, and chain rule are used to differentiate
• Geometrically, a derivative represents the slope of a tangent line

Applications of Derivatives

• Derivatives are used to find maxima and minima in optimization problems
• Derivatives model motion along a line, curve, and surface
• Related rates are used to find rates of change in related quantities

Integrals

• A definite integral is the accumulation of a function over an interval
• The Fundamental Theorem of Calculus relates derivatives and integrals
• Substitution and integration by parts are methods for evaluating integrals

Applications of Integrals

• Integrals are used to find areas under curves
• Integrals are used to find volumes of solids with known cross-sections
• Integrals are used to calculate work and energy in physical systems

Algebra

Equations and Inequalities

• Linear equations are of the form ax + by = c
• Quadratic equations are of the form ax^2 + bx + c = 0
• Systems of linear equations can be solved using substitution, elimination, and matrices

Functions

• A function has a domain and range
• Functions can be composed using the output of one function as the input of another
• Inverse functions can be found using function compositions

Graph Theory

• Graphs consist of vertices, edges, and graphs
• Types of graphs include simple, weighted, directed, and undirected graphs
• Graph algorithms are used for traversing, searching, and sorting graphs

Linear Algebra

• Vectors have operations, properties, and applications
• Matrices have operations, properties, and applications
• Linear transformations are represented using matrices and have applications