Key Concepts from Lecture: Math I (Number Systems to Trigonometry)

Fifty essential vocabulary terms summarizing core definitions, theorems and properties spanning number systems, algebra, geometry, combinatorics, and trigonometry as covered in the lecture notes.

Modulus of a Complex Number

The distance of a complex number a + bi from the origin, calculated as √(a² + b²).

Closure Property

A set is closed under an operation if performing that operation on members of the set always produces a member of the same set.

Transitive Property of Inequality

If a > b and b > c, then a > c.

Irrational Number

A real number that cannot be expressed as the ratio p/q with integers p, q and q ≠ 0.

Multiplicative Inverse

For a non-zero number a, another number 1/a such that a·(1/a) = 1.

Conjugate (Complex)

For z = a + bi, its conjugate is a − bi; their product is a² + b².

Trichotomy Property

For any real number a, exactly one of the relations a>0, a=0, a<0 is true.

Cartesian Product

The set of all ordered pairs formed from two sets A and B, written A × B.

Power Set

The set of all subsets of a given set, including the empty set and the set itself.

Monoid

A semigroup that contains an identity element with respect to the operation.

Group

A set with an associative binary operation, an identity, and inverses for all elements.

Identity Element

An element e such that a∘e = e∘a = a for every element a in the set.

Onto (Surjective) Function

A mapping f: A→B where every element of B is the image of at least one element of A.

One-to-One (Injective) Function

A function where distinct inputs produce distinct outputs.

Bijective Function

A function that is both injective and surjective; it has an inverse function.

Universal Set

The set that contains all objects under discussion, denoted by U.

Binomial Theorem

Provides expansion of (a + b)^n as Σ nCr a^{n-r} b^r for integer n ≥ 0.

Circumcircle

A circle that passes through all three vertices of a triangle.

In-circle

A circle inscribed in a triangle, tangent to each side.

Circumradius (R)

The radius of the circumcircle of a triangle; R = abc / (4Δ).

Inradius (r)

Radius of the inscribed circle; r = Δ / S, where S is the semiperimeter.

Exradius (r₁, r₂, r₃)

Radius of a circle tangent to one side of the triangle and the extensions of the other two sides.

Hero’s Formula

Area of a triangle: Δ = √[S(S − a)(S − b)(S − c)], with S the semiperimeter.

Law of Sines

a/sinα = b/sinβ = c/sinγ = 2R in any triangle.

Law of Cosines

a² = b² + c² − 2bc cosα (and cyclic variants) for any triangle.

Law of Tangents

(a − b)/(a + b) = tan[(α − β)/2] / tan[(α + β)/2].

Periodic Function

A function f satisfying f(x + P) = f(x) for all x and some positive P (the period).

Co-terminal Angles

Angles differing by integer multiples of 2π (or 360°); they share the same terminal side.

Reference Angle

The acute angle formed by the terminal side of an angle and the x-axis.

Principal Value

The unique value of an inverse trig function returned within its principal range.

Radian

Angle subtending an arc equal in length to the radius; 2π radians = 360°.

Degree Measure

Traditional angle unit where a full rotation equals 360 degrees.

Sector of a Circle

Region bounded by two radii and the arc between them; area = ½ r²θ.

Arc Length

For angle θ (radians) in a circle of radius r: s = rθ.

Area by Sine Rule

Δ = ½ ab sin γ = ½ bc sin α = ½ ca sin β.

Pythagorean Triple

Set of three positive integers (a,b,c) satisfying a² + b² = c².

Equilateral Triangle

Triangle with all sides equal and each angle equal to 60°.

Harmonic Mean (H)

For positive numbers a, b: H = 2ab / (a + b).

Geometric Mean (G)

For positive numbers a, b: G = √(ab).

Arithmetic Mean (A)

For numbers a, b: A = (a + b) / 2.

Central Angle

An angle whose vertex is at the center of a circle and whose sides intersect the circle.

Fundamental Trig Identity

sin²x + cos²x = 1 for all real x.

Permutation

An ordered arrangement; nPr = n! / (n − r)!.

Combination

An unordered selection; nCr = n! / [r!(n − r)!].

Factorial (n!)

Product of all positive integers up to n; defined as 1 when n = 0.

Probability

For equally likely outcomes, P(E) = n(E) / n(S).

Sample Space (S)

Set of all possible outcomes of an experiment.

Complement of an Event

Event Ē consisting of all outcomes in S that are not in E; P(Ē) = 1 − P(E).

Even Function

Function satisfying f(−x) = f(x); its graph is symmetric about the y-axis.

Quadrantal Angle

An angle whose terminal side lies on an axis, e.g., 0°, 90°, 180°, 270°.

Unit Circle

Circle of radius 1 centered at the origin; used to define trigonometric functions.