Trigonometry: Angles and Triangles

Angle measurements: Degrees, radians, and conversions between them
Triangles: Right-angled triangles, oblique triangles, and Pythagorean triples
Angle relationships: Complementary, supplementary, and corresponding angles

Angle measurements can be expressed in degrees or radians, with 1 radian equal to 180/π degrees.
Right-angled triangles have one right angle (90°), and oblique triangles have no right angles.
Pythagorean triples are sets of integers that satisfy the Pythagorean theorem, such as 3-4-5.

Sine (sin) is the ratio of the opposite side to the hypotenuse in a right-angled triangle.
Cosine (cos) is the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
Tangent (tan) is the ratio of the opposite side to the adjacent side in a right-angled triangle.
Cotangent (cot) is the ratio of the adjacent side to the opposite side in a right-angled triangle.
Secant (sec) is the ratio of the hypotenuse to the adjacent side in a right-angled triangle.
Cosecant (csc) is the ratio of the hypotenuse to the opposite side in a right-angled triangle.

The Pythagorean identity states that sin²(A) + cos²(A) = 1 for any angle A.
The sum and difference formulas for sine and cosine allow for the calculation of trigonometric ratios for sums and differences of angles.
The double and half angle formulas for sine and cosine allow for the calculation of trigonometric ratios for double and half angles.

Sine and cosine waves have a period of 360° (or 2π radians), and can be shifted horizontally or vertically.
The amplitude of a sine or cosine wave is the maximum displacement from the mean position.
The phase shift of a sine or cosine wave is the horizontal displacement from the origin.
Tangent and cotangent graphs have vertical asymptotes at certain angles.

Trigonometry is used to solve triangles, including finding unknown sides and angles.
Trigonometry is used to model periodic phenomena, such as sound waves and light waves.
Trigonometry is used in analytic geometry to describe coordinates and geometric shapes.
Trigonometry is used in physics and engineering to describe motion, forces, and energies.

Natural numbers are also known as counting numbers, and they start from 1 (e.g. 1, 2, 3,...)
Whole numbers include natural numbers and zero (e.g. 0, 1, 2, 3,...)
Integers include whole numbers and negative numbers (e.g. ..., -3, -2, -1, 0, 1, 2, 3,...)
Rational numbers can be expressed as a fraction (e.g. 1/2, 3/4), and they can be either positive or negative
Irrational numbers cannot be expressed as a fraction (e.g. π, e), and they are non-repeating and non-terminating
Real numbers include both rational and irrational numbers, making them a comprehensive set of numbers
Complex numbers include real numbers and imaginary numbers (e.g. 3 + 4i), and they are used to represent quantities with both magnitude and direction

Variables are letters or symbols that represent unknown values, and they are used to solve equations and inequalities
Constants are numbers that do not change, and they are used to perform mathematical operations
Expressions are combinations of variables, constants, and mathematical operations, and they can be simplified or evaluated
Equations are statements that express the equality of two expressions, and they can be solved using algebraic methods
Inequalities are statements that compare two expressions using greater than, less than, or equal to, and they can be solved using algebraic methods

Points are locations in space, represented by coordinates (x, y), and they can be used to define shapes and figures
Lines are sets of points that extend infinitely in two directions, and they can be used to define shapes and figures
Angles are formed by two lines or rays sharing a common endpoint, and they can be acute, obtuse, or right angles
Triangles are polygons with three sides and three angles, and they can be classified into different types (e.g. equilateral, isosceles, scalene)
Quadrilaterals are polygons with four sides and four angles, and they can be classified into different types (e.g. rectangles, squares, trapezoids)

Units are standard amounts used to measure quantities, such as inches, meters, and grams
Conversion involves changing from one unit to another, such as from inches to feet or from pounds to kilograms
Perimeter is the distance around a shape, and it can be calculated using different formulas depending on the shape
Area is the size of a two-dimensional shape, and it can be calculated using different formulas depending on the shape
Volume is the amount of space inside a three-dimensional shape, and it can be calculated using different formulas depending on the shape

Limits involve the behavior of a function as the input approaches a certain value, and they are used to define the basic operations of calculus
Derivatives are rates of change of functions, and they are used to find the slope of a tangent line to a curve at a point
Integrals are accumulations of quantities over a defined interval, and they are used to find the area under a curve or the volume of a solid
Applications of calculus involve optimization, motion, and related rates, and they are used to solve real-world problems in fields such as physics, engineering, and economics

Data is a collection of information, and it can be qualitative or quantitative
Graphs are visual representations of data, and they can be used to display and analyze data
Mean is the average value of a dataset, and it can be calculated using different methods depending on the dataset
Median is the middle value of a dataset when in order, and it can be used to describe the central tendency of a dataset
Mode is the most frequently occurring value in a dataset, and it can be used to describe the central tendency of a dataset
Probability is the chance of an event occurring, and it can be calculated using different methods depending on the event and the dataset