Trigonometry, Geometry, and Calculus Overview

Trigonometry

• Definition: Study of relationships between angles and sides of triangles.
• Key Functions:
  • Sine (sin) - opposite/hypotenuse
  • Cosine (cos) - adjacent/hypotenuse
  • Tangent (tan) - opposite/adjacent
• Pythagorean Identity: ( sin^2(x) + cos^2(x) = 1 )
• Important Angles: 0°, 30°, 45°, 60°, 90° - with corresponding sin, cos, tan values.
• Unit Circle: Circle of radius 1 used to define trigonometric functions.

Geometry

• Definition: Study of shapes, sizes, and properties of space.
• Basic Shapes:
  • Triangle: Sum of angles = 180°
  • Quadrilateral: Sum of angles = 360°
  • Circle: Area = ( \pi r^2 ), Circumference = ( 2\pi r )
• Theorems:
  • Pythagorean Theorem: ( a^2 + b^2 = c^2 ) for right triangles.
  • Area formulas for common shapes (e.g., rectangle: base × height).
• Congruence and Similarity: Criteria for triangles (SSS, SAS, ASA).

Calculus

• Definition: Study of change and motion; involves derivatives and integrals.
• Key Concepts:
  • Derivative: Measures rate of change; ( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} )
  • Integral: Represents area under a curve; Fundamental Theorem connects differentiation and integration.
• Types of Integrals:
  • Definite Integrals: Calculate area; has upper and lower limits.
  • Indefinite Integrals: General form; no limits, includes constant of integration (C).
• Applications: Motion, optimization, area under curves.

Algebra

• Definition: Study of mathematical symbols and rules for manipulating those symbols.
• Key Concepts:
  • Variables: Symbols that represent numbers (e.g., x, y).
  • Expressions: Combinations of numbers and variables (e.g., ( 2x + 3 )).
  • Equations: Statements that two expressions are equal (e.g., ( 2x + 3 = 7 )).
• Operations:
  • Solving linear equations: Isolate variable.
  • Factoring polynomials: Rewrite expressions as products (e.g., ( x^2 - 5x + 6 = (x-2)(x-3) )).
• Functions: Relations that assign each input exactly one output.

Statistics

• Definition: Study of data collection, analysis, interpretation, presentation, and organization.
• Key Concepts:
  • Descriptive Statistics: Summarizes data (mean, median, mode).
  • Inferential Statistics: Makes predictions/inferences about a population based on a sample.
• Common Distributions:
  • Normal Distribution: Bell-shaped curve; mean = median = mode.
  • Binomial Distribution: Describes number of successes in a fixed number of trials.
• Hypothesis Testing: Procedure to test assumptions about population parameters (null vs. alternative hypothesis).
• Correlation vs. Causation: Correlation measures relationship between variables, while causation indicates one variable affects another.

Trigonometry

• Definition focuses on the relationships between angles and sides in triangles.
• Key trigonometric functions:
  • Sine (sin) computes as opposite side divided by hypotenuse.
  • Cosine (cos) computes as adjacent side divided by hypotenuse.
  • Tangent (tan) computes as opposite side divided by adjacent side.
• Pythagorean Identity shows the relationship: ( sin^2(x) + cos^2(x) = 1 ).
• Important angles with corresponding function values:
  • 0°: sin = 0, cos = 1, tan = 0
  • 30°: sin = 0.5, cos = ( \sqrt{3}/2 ), tan = ( 1/\sqrt{3} )
  • 45°: sin = cos = ( \sqrt{2}/2 ), tan = 1
  • 60°: sin = ( \sqrt{3}/2 ), cos = 0.5, tan = ( \sqrt{3} )
  • 90°: sin = 1, cos = 0, tan = undefined
• The unit circle facilitates the understanding of trigonometric functions and their values.

Geometry

• Definition encompasses the study of shapes, sizes, and spatial properties.
• Basic shapes and their angle sums:
  • Triangle: Total of interior angles equals 180°.
  • Quadrilateral: Total of interior angles equals 360°.
  • Circle: Area calculated as ( \pi r^2 ) and circumference as ( 2\pi r ).
• Pythagorean Theorem establishes the relationship in right triangles: ( a^2 + b^2 = c^2 ).
• Area formulas vary by shape; for example, the area of a rectangle is found by multiplying base and height.
• Congruence and similarity are evaluated through criteria such as Side-Side-Side (SSS), Side-Angle-Side (SAS), and Angle-Side-Angle (ASA).

Calculus

• Definition emphasizes the analysis of change and motion via derivatives and integrals.
• Key concepts inform about:
  • Derivative measuring the rate of change, formulated as ( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ).
  • Integral representing the area under a curve, linked to differentiation by the Fundamental Theorem of Calculus.
• Types of integrals include:
  • Definite integrals with specific upper and lower limits, used to calculate precise areas.
  • Indefinite integrals do not have limits and include a constant of integration (C).
• Applications of calculus span areas like motion, optimization problems, and calculating areas under curves.

Algebra

• Definition addresses the manipulation of mathematical symbols and formulas.
• Key concepts explain:
  • Variables represent unknowns, commonly denoted as x or y.
  • Expressions consist of numbers and variables combined through operations (e.g., ( 2x + 3 )).
  • Equations express equality between two algebraic expressions (e.g., ( 2x + 3 = 7 )).
• Operations include:
  • Techniques for solving linear equations aim to isolate the variable.
  • Factoring polynomials involves rewriting expressions into products (e.g., ( x^2 - 5x + 6 = (x-2)(x-3) )).
• Functions are defined as relations that assign exactly one output for each input.

Statistics

• Definition focuses on the methodologies for data collection, analysis, interpretation, and presentation.
• Key concepts include:
  • Descriptive statistics provide summaries of data such as mean, median, and mode.
  • Inferential statistics allow for predictions or conclusions about a population based on a sample.
• Common statistical distributions:
  • Normal distribution is characterized by a bell-shaped curve where mean, median, and mode are equal.
  • Binomial distribution evaluates the number of successes in a fixed number of independent trials.
• Hypothesis testing involves procedures for validating assumptions about population parameters, comparing null and alternative hypotheses.
• Distinct differences between correlation, which assesses relationships between variables, and causation, which indicates a direct effect of one variable on another.

This quiz covers fundamental concepts in Trigonometry, Geometry, and Calculus. It includes definitions, key functions, theorems, and properties of shapes and angles. Perfect for students looking to reinforce their understanding of these essential math topics.