Comprehensive Study Notes: College Algebra and Trigonometry (MATH1034A)

Numbers, Inequalities, and Absolute Values

Sets

Sets are fundamental objects in mathematics, representing collections of distinct elements. Understanding set notation and operations is essential for algebraic reasoning.

• Set: A well-defined collection of objects, called elements.

• Notation: Curly braces are used, e.g., {1, 2, 3}.

• Set-builder notation: {x | A(x)} denotes the set of all x for which the statement A(x) is true.

• Subset: A ⊂ B means every element of A is also in B.

• Intersection: A ∩ B = {x | x ∈ A and x ∈ B}.

• Union: A ∪ B = {x | x ∈ A or x ∈ B}.

• Empty set: The set with no elements, denoted ∅.

Example: The set of all continents: {Africa, Antarctica, Australia, Asia, Europe, North America, South America}.

Real Numbers

The real number system is built from several subsets:

• Natural numbers (N): {1, 2, 3, ...}

• Integers (Z): {..., -2, -1, 0, 1, 2, ...}

• Rational numbers (Q): Numbers expressible as p/q where p, q ∈ Z, q ≠ 0.

• Irrational numbers: Real numbers not rational, e.g., √2, π.

• Real numbers (R): All points on the number line, including both rationals and irrationals.

Ordering: For a, b ∈ R, a < b means a is to the left of b on the number line. Notations: a ≤ b (a less than or equal to b), a ≥ b (a greater than or equal to b).

Intervals

Intervals are subsets of R corresponding to line segments:

• Open interval: (a, b) = {x ∈ R | a < x < b}

• Closed interval: [a, b] = {x ∈ R | a ≤ x ≤ b}

• Half-open intervals: [a, b) or (a, b]

• Unbounded intervals: (a, ∞), (-∞, b)

Inequalities

Solving inequalities involves finding all real numbers satisfying a given condition.

• Example: Solve 3x + 1 > 2x:

  • Subtract 2x: x + 1 > 0

  • Subtract 1: x > -1

  • Solution: (-1, ∞)

• Compound inequalities: Use logical connectors (AND/OR) to combine conditions.

• Express solutions: In set or interval notation.

Absolute Value

The absolute value of a real number a, denoted |a|, is its distance from zero on the number line.

• Definition: $|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$

• Key properties:

  • $|a| \geq 0$ for all a

  • $|a| = 0$ if and only if $a = 0$

  • $|a|^2 = a^2$

  • $|ab| = |a||b|$

  • $|a + b| \leq |a| + |b|$ (Triangle Inequality)

  • $|a| = \sqrt{a^2}$

• Solving absolute value equations/inequalities:

  • $|x| = r \implies x = r$ or $x = -r$

  • $|x| < r \implies -r < x < r$

  • $|x| > r \implies x < -r$ or $x > r$

Functions

Definition and Basics

A function f from set D to set Y assigns a unique element f(x) ∈ Y to each x ∈ D. D is the domain, and the set of all f(x) is the range.

• Notation: $f: D \to Y$, $y = f(x)$

• Natural domain: All x for which f(x) is defined as a real number.

Example: $f(x) = \frac{1}{\sqrt{x-1}}$ has domain $(1, \infty)$.

Graphing Functions

• Graph: The set of points $(x, f(x))$ for $x \in D$.

• Vertical Line Test: A curve is the graph of a function if no vertical line intersects it more than once.

• Piecewise functions: Defined by different formulas on different parts of the domain.

• Increasing/Decreasing: f is increasing if $f(x_1) < f(x_2)$ for $x_1 < x_2$; decreasing if $f(x_1) > f(x_2)$.

Transformations

• Shifts: $y = f(x + c)$ (horizontal), $y = f(x) + c$ (vertical)

• Scaling: $y = f(cx)$ (horizontal), $y = cf(x)$ (vertical)

• Reflections: $y = f(-x)$ (about y-axis), $y = -f(x)$ (about x-axis)

Even and Odd Functions

• Even: $f(-x) = f(x)$ for all $x$ in domain (symmetric about y-axis)

• Odd: $f(-x) = -f(x)$ for all $x$ in domain (symmetric about origin)

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Classification and Combination of Functions

• Polynomial: $p(x) = a_nx^n + ... + a_0$

• Rational: $g(x) = \frac{p(x)}{q(x)}$ where p, q are polynomials, $q(x) \neq 0$

• Algebraic: Built from polynomials using roots and rational operations

• Operations: $(f+g)(x) = f(x) + g(x)$, $(fg)(x) = f(x)g(x)$, $(f/g)(x) = \frac{f(x)}{g(x)}$ (where $g(x) \neq 0$)

• Composite: $(f \circ g)(x) = f(g(x))$

Inverse Functions

• One-to-one (injective): $f(x_1) = f(x_2) \implies x_1 = x_2$

• Inverse: If $f$ is one-to-one, $f^{-1}(y) = x$ such that $f(x) = y$

• Horizontal Line Test: A function is one-to-one if every horizontal line intersects its graph at most once.

• Graph of inverse: Reflection of the graph of $f$ across the line $y = x$

Angles and Trigonometric Functions

Radian Measure

• Definition: The radian measure of an angle is the ratio of arc length $s$ to radius $r$: $\alpha = \frac{s}{r}$

• Conversion: $1$radian$= \frac{180}{\pi}$degrees; $1$ degree $= \frac{\pi}{180}$ radians

• Arc length: $s = r\alpha$ (with $\alpha$ in radians)

• Area of sector: $A = \frac{1}{2} r^2 \alpha$

Trigonometric Functions

• Definitions (unit circle):

  • $\sin \theta = \frac{y}{r}$

  • $\cos \theta = \frac{x}{r}$

  • $\tan \theta = \frac{y}{x}$

  • $\csc \theta = \frac{r}{y}$

  • $\sec \theta = \frac{r}{x}$

  • $\cot \theta = \frac{x}{y}$

• Special angles: Know exact values for $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$

• Periodicity: $\sin$ and $\cos$ have period $2\pi$; $\tan$ and $\cot$ have period $\pi$

Trigonometric Identities

• Pythagorean: $\sin^2 \theta + \cos^2 \theta = 1$

• Quotient: $\tan \theta = \frac{\sin \theta}{\cos \theta}$, $\cot \theta = \frac{\cos \theta}{\sin \theta}$

• Reciprocal: $\sec \theta = \frac{1}{\cos \theta}$, $\csc \theta = \frac{1}{\sin \theta}$

• Sum and difference: $\sin(\theta \pm \phi) = \sin \theta \cos \phi \pm \cos \theta \sin \phi$

• Double angle: $\sin(2\theta) = 2\sin \theta \cos \theta$

• Product-to-sum and sum-to-product formulas

Inverse Trigonometric Functions

• arcsin: $y = \arcsin x$ is the unique $y \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $\sin y = x$

• arccos: $y = \arccos x$ is the unique $y \in [0, \pi]$ with $\cos y = x$

• arctan: $y = \arctan x$ is the unique $y \in (-\frac{\pi}{2}, \frac{\pi}{2})$ with $\tan y = x$

• Domains and ranges: Know the principal values for each function

Trigonometric Equations

• General solutions:

  • $\sin \theta = x \implies \theta = \arcsin x + 2k\pi$ or $\theta = \pi - \arcsin x + 2k\pi$

  • $\cos \theta = x \implies \theta = \arccos x + 2k\pi$ or $\theta = -\arccos x + 2k\pi$

  • $\tan \theta = x \implies \theta = \arctan x + k\pi$

Polar Coordinates

• Definition: A point in the plane is given by $(r, \theta)$, where $r \geq 0$ is the distance from the origin and $\theta$ is the angle from the positive x-axis.

• Conversion:

  • $x = r \cos \theta$

  • $y = r \sin \theta$

  • $r = \sqrt{x^2 + y^2}$

  • $\tan \theta = \frac{y}{x}$ (with quadrant considerations)

• Polar equations: Equations in $r$ and $\theta$ can describe circles, lines, spirals, roses, cardioids, etc.

• Symmetry tests: Replace $\theta$ with $-\theta$, $\pi - \theta$, or $\pi + \theta$ to test for symmetry about x-axis, y-axis, or origin.

Expressing $a \cos x + b \sin x$ as $R \cos(x - \theta)$

• Any expression of the form $a \cos x + b \sin x$ can be rewritten as $R \cos(x - \theta)$, where $R = \sqrt{a^2 + b^2}$ and $\tan \theta = \frac{b}{a}$.

• Example: $\cos x + \sqrt{3} \sin x = 2 \cos(x - \frac{\pi}{3})$

Mathematical Induction

Principle of Mathematical Induction

• To prove a statement $P(n)$ for all $n \in \mathbb{N}$:

  1. Show $P(1)$ is true (base case).

  2. Assume $P(k)$ is true; show $P(k+1)$ is true (inductive step).

• If both steps are satisfied, $P(n)$ is true for all $n \in \mathbb{N}$.

Example: Prove $1 + 2 + ... + n = \frac{n(n+1)}{2}$ for all $n \in \mathbb{N}$.

Sigma Notation and Binomial Theorem

Sigma Notation

• Definition: $\sum_{j=1}^n a_j = a_1 + a_2 + ... + a_n$

• Properties: Linearity, index shifting, splitting sums, etc.

Summation Formulas

• Arithmetic series: $\sum_{j=0}^{n-1} (a + jd) = \frac{n}{2}[2a + (n-1)d]$

• Geometric series: $\sum_{j=0}^{n-1} ar^j = a \frac{1 - r^n}{1 - r}$ (for $r \neq 1$)

• Power sums:

  • $\sum_{r=1}^n r = \frac{n(n+1)}{2}$

  • $\sum_{r=1}^n r^2 = \frac{n(n+1)(2n+1)}{6}$

  • $\sum_{r=1}^n r^3 = \frac{n^2(n+1)^2}{4}$

Factorials and Binomial Coefficients

• Factorial: $n! = 1 \cdot 2 \cdot ... \cdot n$, $0! = 1$

• Binomial coefficient: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

• Properties: $\binom{n}{k} = \binom{n}{n-k}$, $\binom{n}{0} = 1$, $\binom{n}{1} = n$

• Pascal's Rule: $\binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}$

Binomial Theorem

• Statement: $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$

• Applications: Expanding powers, finding coefficients, constant terms, etc.

• Example: Expand $(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

Conic Sections

Quadratic Forms and Canonical Forms

• General quadratic: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

• Canonical forms:

  • Parabola: $y^2 = 4ax$

  • Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

  • Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

• Classification: By completing the square and, if necessary, rotating axes to eliminate the $xy$ term.

• Ellipse: Both $x^2$ and $y^2$ terms have the same sign.

• Hyperbola: $x^2$ and $y^2$ terms have opposite signs.

• Parabola: Only one variable is squared.

Change of Axes

• Translation: Shifting the origin to simplify the equation.

• Rotation: Used to eliminate the $xy$ term. The angle $\alpha$ is found by $\cot 2\alpha = \frac{A - C}{B}$.

• Principal axes: The new axes after rotation, along which the conic is in canonical form.

Appendix: Mathematical Reasoning and Notation

• Statements: Expressions that are either true or false.

• Implication: "If p, then q" ($p \Rightarrow q$)

• Converse: $q \Rightarrow p$

• Contrapositive: $\neg q \Rightarrow \neg p$ (logically equivalent to the original implication)

• Equivalence: $p \Leftrightarrow q$ (both $p \Rightarrow q$ and $q \Rightarrow p$ are true)

• Proof methods: Direct, indirect (contradiction), and by induction.

Table: Types of Intervals

Notation	Set Description	Type
(a, b)	{x ∈ R | a < x < b}	Open
[a, b]	{x ∈ R | a ≤ x ≤ b}	Closed
[a, b)	{x ∈ R | a ≤ x < b}	Half-open
(a, b]	{x ∈ R | a < x ≤ b}	Half-open
(a, ∞)	{x ∈ R | x > a}	Open
[a, ∞)	{x ∈ R | x ≥ a}	Closed
(-∞, b)	{x ∈ R | x < b}	Open
(-∞, b]	{x ∈ R | x ≤ b}	Closed
(-∞, ∞)	R	Both open and closed

Additional info: Some advanced topics (e.g., rotation of axes, principal axes, and the completeness axiom) are included for completeness and context, as they are foundational for further study in algebra and analytic geometry.