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Questions and Answers
What is the value of sin(30°) in a right triangle?
What is the derivative of the function f(x) = 3x^2?
What is the equation of the line tangent to the curve y = x^2 at the point (2, 4)?
What is the value of $\int\limits_{0}^{1} x^2 dx$?
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What is the solution to the equation x^2 + 4x + 4 = 0?
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What is the value of cos(60°) in a right triangle?
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What is the formula for the sum of the angles in a triangle?
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What is the derivative of the function f(x) = sin(x)?
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What is the equation of the line normal to the curve y = x^2 at the point (2, 4)?
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What is the value of $\int\limits_{0}^{2} 2x dx$?
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Study Notes
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
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Description
Test your understanding of angles, triangles, and trigonometric ratios. Learn about properties of triangles, sine, cosine, and tangent functions, and identify trigonometric identities and formulas.
