Trigonometric functions

Algebraic Equations

Definition

• An algebraic equation is an equation involving variables and constants, where the variables are raised to a power (usually a whole number) and combined using addition, subtraction, multiplication, and division.

Types of Algebraic Equations

• Linear Equations: degree of the variable(s) is 1
  • Example: 2x + 3 = 5
• Quadratic Equations: degree of the variable(s) is 2
  • Example: x^2 + 4x + 4 = 0
• Polynomial Equations: degree of the variable(s) is 3 or higher
  • Example: x^3 - 2x^2 + x - 1 = 0

Properties of Algebraic Equations

• Commutative Property: the order of variables and constants does not change the equation
• Associative Property: the order in which operations are performed does not change the equation
• Distributive Property: multiplication distributes over addition

Solving Algebraic Equations

• Addition and Subtraction: add or subtract the same value to both sides of the equation
• Multiplication and Division: multiply or divide both sides of the equation by the same non-zero value
• Factoring: express the equation as a product of simpler equations
• Quadratic Formula: used to solve quadratic equations of the form ax^2 + bx + c = 0

Applications of Algebraic Equations

• Modeling Real-World Problems: algebraic equations can be used to model population growth, electrical circuits, and other real-world phenomena
• Science and Engineering: algebraic equations are used to describe the laws of physics, chemistry, and other scientific fields

Algebraic Equations

Definition

• Involves variables and constants combined using addition, subtraction, multiplication, and division
• Variables are raised to a power (usually a whole number)

Types of Algebraic Equations

Linear Equations

• Degree of the variable(s) is 1
• Example: 2x + 3 = 5

Quadratic Equations

• Degree of the variable(s) is 2
• Example: x^2 + 4x + 4 = 0

Polynomial Equations

• Degree of the variable(s) is 3 or higher
• Example: x^3 - 2x^2 + x - 1 = 0

Properties of Algebraic Equations

Commutative Property

• Order of variables and constants does not change the equation

Associative Property

• Order in which operations are performed does not change the equation

Distributive Property

• Multiplication distributes over addition

Solving Algebraic Equations

Elementary Operations

• Add or subtract the same value to both sides of the equation
• Multiply or divide both sides of the equation by the same non-zero value

Factoring

• Express the equation as a product of simpler equations

Quadratic Formula

• Used to solve quadratic equations of the form ax^2 + bx + c = 0

Applications of Algebraic Equations

Modeling Real-World Problems

• Algebraic equations can be used to model population growth, electrical circuits, and other real-world phenomena

Science and Engineering

• Algebraic equations are used to describe the laws of physics, chemistry, and other scientific fields

Trigonometric Functions

Definition and Basics

• Trigonometric functions are based on the ratios of sides in right-angled triangles.
• There are three basic trigonometric functions: sine (sin), cosine (cos), and tangent (tan), which are defined as:
  • Sine (sin): opposite side / hypotenuse
  • Cosine (cos): adjacent side / hypotenuse
  • Tangent (tan): opposite side / adjacent side

Identities and Formulas

• The Pythagorean Identity states that sin²(x) + cos²(x) = 1.
• The Sum and Difference Formulas are:
  • sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
  • cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
  • tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))
• The Double Angle Formulas are:
  • sin(2x) = 2sin(x)cos(x)
  • cos(2x) = cos²(x) - sin²(x)
  • tan(2x) = (2tan(x)) / (1 - tan²(x))

Graphs and Properties

• Trigonometric functions have a periodic nature, repeating every 360° (or 2π radians).
• The amplitude is the maximum value of a trigonometric function.
• A phase shift occurs when a trigonometric function is horizontally shifted.

Applications

• Triangulation is used to find unknown sides and angles in triangles.
• Trigonometric functions model real-world phenomena, such as sound waves, light waves, and electrical signals.
• Analytic geometry uses trigonometry to define curves and surfaces.