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Questions and Answers
Which property of algebraic equations states that the order of variables and constants does not change the equation?
Which property of algebraic equations states that the order of variables and constants does not change the equation?
What is the degree of the variable in a linear equation?
What is the degree of the variable in a linear equation?
What is the method used to solve quadratic equations of the form ax^2 + bx + c = 0?
What is the method used to solve quadratic equations of the form ax^2 + bx + c = 0?
What type of equation has a degree of the variable(s) of 3 or higher?
What type of equation has a degree of the variable(s) of 3 or higher?
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Which of the following is an application of algebraic equations?
Which of the following is an application of algebraic equations?
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What is the purpose of the Distributive Property in solving algebraic equations?
What is the purpose of the Distributive Property in solving algebraic equations?
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What is the ratio of the opposite side to the hypotenuse in a right-angled triangle?
What is the ratio of the opposite side to the hypotenuse in a right-angled triangle?
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What is the Pythagorean Identity in trigonometry?
What is the Pythagorean Identity in trigonometry?
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Which of the following formulas is used to find the sine of the sum of two angles?
Which of the following formulas is used to find the sine of the sum of two angles?
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What is the period of a trigonometric function?
What is the period of a trigonometric function?
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What is the application of trigonometry in finding unknown sides and angles in triangles?
What is the application of trigonometry in finding unknown sides and angles in triangles?
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What is the double angle formula for cosine?
What is the double angle formula for cosine?
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Study Notes
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.
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