Algebra and Trigonometry Lecture Notes

Vocabulary flashcards covering basic properties of real numbers, algebra, coordinate geometry, and trigonometry from the lecture transcript.

Real Numbers (ℝ)

The set of numbers encompassing all integers, rational, and irrational numbers, which is closed under addition and multiplication.

Additive Inverse

A unique number -x for any real number x such that x + (-x) = 0.

Multiplicative Inverse

For any non-zero real number x, a unique number x⁻¹ such that x · x⁻¹ = 1.

Distributive Law

The property stating that for all real numbers x, y, and z, x · (y + z) = x · y + x · z.

Zero Factor Property

The property stating that x · y = 0 if and only if x = 0 or y = 0.

Rational Number (ℚ)

A real number that can be expressed as a ratio or division of two integers.

Irrational Number

A real number that cannot be expressed as a ratio of two integers, such as √2.

Linear Equation

An equation of the form ax + b = 0, representing a straight line.

Equivalent Equations

Equations that have the same solution set.

Quadratic Equation

An equation of the form ax² + bx + c = 0, where a is non-zero.

Discriminant (Δ)

The expression b² - 4ac used to determine the number and nature of roots in a quadratic equation.

Polynomial

An expression consisting of variables and coefficients in the form an xⁿ + an₋₁xⁿ⁻¹ + … + a₀.

Monomial

An expression of the form xᵏ where k is a non-negative integer.

Zero Polynomial

A polynomial in which all coefficients are equal to zero.

Division Algorithm

The theorem stating P(x) = K(x)Q(x) + R(x), where the remainder R(x) has a lower degree than the divisor.

Remainder Theorem

States that the remainder of dividing polynomial P(x) by (x - a) is equal to P(a).

Factor Theorem

States that (x - a) is a factor of P(x) if and only if P(a) = 0.

Rational Root Theorem

Relates the potential rational solutions m/k of a polynomial to its leading coefficient and constant term.

Complex Numbers (ℂ)

Numbers of the form a + bi, where a and b are real numbers and i = √-1.

Conjugate

The complex number a - bi for a given complex number a + bi.

Function

A correspondence between sets A and B that assigns to each element x in A exactly one element f(x) in B.

Domain

The set of all possible input values for a function.

Range

The set of all output values produced by a function.

One-to-One Function

A function where distinct inputs always correspond to distinct outputs.

Inverse Function (f⁻¹)

A unique function such that y = f(x) if and only if x = f⁻¹(y).

y-intercept

The point (0, b) where a graph intersects the vertical axis.

Parabola

The characteristic U-shaped graph of a quadratic function.

Vertex

The minimum or maximum point on a parabola.

Exponential Function

A function of the form y = aˣ where a > 0.

Logarithmic Function

The inverse of an exponential function, written y = logₐ x.

Natural Logarithm (ln x)

A logarithm with the irrational base e (approximately 2.718).

Rational Expression

A fraction where both the numerator and denominator are polynomials.

Asymptote

A line that a graph approaches but never intersects.

Pythagorean Theorem

In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse (a² + b² = c²).

Distance Formula

The formula d = √((x₁ - x₂)² + (y₁ - y₂)²) used to calculate the space between two points.

Slope (m)

The measure of steepness of a line, calculated as rise over run: (y₂ - y₁) / (x₂ - x₁).

Slope-Intercept Form

The form y = mx + d for a linear equation.

Perpendicular Bisector

A line passing through the midpoint of a segment at a 90-degree angle.

Sine (sin α)

The ratio of the y-coordinate to the radius (y/r) for an angle in a coordinate system.

Tangent (tan α)

The ratio of sine to cosine (y/x), representing the slope of an angle's terminal side.

Cosine (cos α)

Defined for an acute angle as sin(90° - α), or the ratio x/r.

Cotangent (cot α)

The reciprocal of the tangent function, defined as x/y.

Secant (sec α)

The reciprocal of the cosine function (1/cos α).

Cosecant (csc α)

The reciprocal of the sine function (1/sin α).

Double-Angle Identities

Trigonometric formulas used to find values for sin 2α, cos 2α, and tan 2α.

Periodic Function

A function that repeats its values in regular intervals or cycles.

Amplitude

The absolute value of the coefficient A in a periodic function y = Af(x).

Law of Sines

The relationship a/sin α = b/sin β = c/sin γ = 2r for any triangle.

Law of Cosines

An extension of the Pythagorean theorem: a² = b² + c² - 2bc cos α.

Radian

A unit of angular measure defined as the ratio of the arc length to the radius (l/r).