Trigonometry and Calculus Overview

Definition: Study of relationships between angles and sides of triangles.
Key Functions:
- Sine (sin)
- Cosine (cos)
- Tangent (tan)
Important Identities:
- Pythagorean Identity: ( \sin^2(x) + \cos^2(x) = 1 )
- Angle Sum and Difference:
  - ( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) )
  - ( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) )
- Double Angle Formulas:
  - ( \sin(2x) = 2\sin(x)\cos(x) )
  - ( \cos(2x) = \cos^2(x) - \sin^2(x) )
Applications: Wave functions, oscillations, engineering, and navigation.

Definition: Study of change and motion; deals with derivatives and integrals.
Fundamental Theorem: Connects differentiation and integration.
Key Concepts:
- Limits: Value that a function approaches as the input approaches a point.
- Derivatives:
  - Measure of how a function changes as its input changes.
  - Rules: Power Rule, Product Rule, Quotient Rule, Chain Rule.
- Integrals:
  - Represents accumulation of quantities and area under curves.
  - Types: Definite (specific limits) and Indefinite (no limits).
Applications: Physics, engineering, economics, and optimization problems.

Definition: Science of collecting, analyzing, interpreting, and presenting data.
Key Concepts:
- Descriptive Statistics: Summarizes data (mean, median, mode, variance, standard deviation).
- Inferential Statistics: Makes predictions or inferences about a population based on a sample.
- Probability: Measures how likely an event is to occur.
Distributions:
- Normal Distribution: Bell-shaped curve; characterized by mean and standard deviation.
- Binomial Distribution: Discrete distribution of the number of successes in a sequence of n trials.
Hypothesis Testing: Process of making decisions using data; involves null and alternative hypotheses, p-values, and significance levels.

Definition: Branch of mathematics concerning shapes, sizes, and properties of space.
Basic Shapes:
- 2D: Circles, triangles, squares, rectangles, polygons.
- 3D: Spheres, cubes, cylinders, cones, pyramids.
Key Concepts:
- Angles: Measured in degrees; types include acute, right, obtuse, and straight.
- Theorems:
  - Pythagorean Theorem: ( a^2 + b^2 = c^2 ) for right triangles.
  - Area and perimeter formulas for various shapes.
  - Volume formulas for 3D shapes.
Applications: Architecture, engineering, computer graphics, and various fields involving spatial reasoning.

Studies the relationships between angles and sides of triangles.
Key trigonometric functions include sine (sin), cosine (cos), and tangent (tan).
Important trigonometric identities:
- Pythagorean Identity: ( \sin^2(x) + \cos^2(x) = 1 )
- Angle Sum and Difference: ( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) ) and ( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) )
- Double Angle Formulas: ( \sin(2x) = 2\sin(x)\cos(x) ) and ( \cos(2x) = \cos^2(x) - \sin^2(x) )
Applications include wave functions, oscillations, engineering, and navigation.

The study of change and motion.
Deals with derivatives and integrals.
The Fundamental Theorem of Calculus connects differentiation and integration.
Key concepts include limits, derivatives, and integrals.
- Limits represent the value a function approaches as its input gets closer to a specific point.
- Derivatives measure how a function changes as its input changes.
- Integrals represent the accumulation of quantities and the area under curves.
- Types of integrals include definite (with specific limits) and indefinite (without limits).
Applications include physics, engineering, economics, and optimization problems.

The science of collecting, analyzing, interpreting, and presenting data.
Key concepts include descriptive statistics, inferential statistics, probability, and distributions.
- Descriptive statistics summarize data using measures like mean, median, mode, variance, and standard deviation.
- Inferential statistics make predictions or inferences about a population based on a sample.
- Probability measures the likelihood of an event occurring.
- Distributions are models of data such as the normal distribution (bell-shaped curve) and the binomial distribution (for discrete successes).
Hypothesis testing involves making decisions using data through null and alternative hypotheses, p-values, and significance levels.

The branch of mathematics focused on shapes, sizes, and properties of space.
Basic shapes include 2D figures like circles, triangles, squares, rectangles, and polygons, and 3D figures like spheres, cubes, cylinders, cones, and pyramids.
Key concepts include angles (acute, right, obtuse, straight) and theorems.
- Pythagorean Theorem states ( a^2 + b^2 = c^2 ) for right triangles.
Applications include architecture, engineering, computer graphics, and various fields requiring spatial reasoning.