College Algebra and Trigonometry: Structured Study Notes

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.

• Definition: A set is a well-defined collection of objects, called elements.

• Notation: Sets are denoted by curly brackets, e.g., A = {1, 2, 3}.

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

• Subset: A ⊂ B if 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: C = {Africa, Antarctica, Australia, Asia, Europe, North America, South America}.

Real Numbers

The real numbers encompass several important subsets:

• Natural numbers (N): $N = \{1, 2, 3, \ldots\}$

• Integers (Z): $Z = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$

• Rational numbers (Q): $Q = \left\{ \frac{p}{q} : p, q \in Z, q \neq 0 \right\}$

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

• Irrational numbers: Real numbers not expressible as a ratio of integers, e.g., $\sqrt{2}, \pi$.

Intervals: Subsets of the real line, e.g.,

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

• Closed interval: $[a, b] = \{x \in R | a \leq x \leq b\}$

Inequalities

Inequalities describe the relative size of real numbers and are solved using algebraic manipulation.

• Symbols: $a < b$, $a \leq b$, $a > b$, $a \geq b$

• Solution sets: Expressed in set or interval notation.

Example: Solve $3x + 1 > 2x$:

• $3x + 1 > 2x \implies x > -1$

• Solution: $x \in (-1, \infty)$

Absolute Values

The absolute value of a real number measures its distance from zero.

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

• Key properties:

  • $|a| \geq 0$

  • $|a|^2 = a^2$

  • $|ab| = |a||b|$

  • $|a + b| \leq |a| + |b|$ (triangle inequality)

  • $|a| = \sqrt{a^2}$

Example: Solve $|2x + 4| < 1$:

• $-1 < 2x + 4 < 1 \implies -5 < 2x < -3 \implies -\frac{5}{2} < x < -\frac{3}{2}$

Functions

Definition and Basics

A function assigns each element in a domain to a unique element in a codomain.

• Definition: $f: D \to Y$ assigns $f(x) \in Y$ to each $x \in D$.

• Domain: The set $D$ of allowable inputs.

• Range: The set $\{f(x) | x \in D\}$ of outputs.

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

Graphing Techniques

• Graph: The set of points $(x, f(x))$ in the plane.

• 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 intervals.

Transformations

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

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

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

Even and Odd Functions

• Even: $f(-x) = f(x)$ (symmetric about y-axis)

• Odd: $f(-x) = -f(x)$ (symmetric about origin)

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

Classification and Combination of Functions

• Polynomial: $p(x) = a_n x^n + \ldots + a_0$

• Rational: $g(x) = \frac{p(x)}{q(x)}$

• Algebraic: Built from polynomials using roots and rational operations.

Operations:

Operation	Definition	Domain
$f + g$	$(f + g)(x) = f(x) + g(x)$	$\text{dom}(f) \cap \text{dom}(g)$
$f - g$	$(f - g)(x) = f(x) - g(x)$	$\text{dom}(f) \cap \text{dom}(g)$
$fg$	$(fg)(x) = f(x)g(x)$	$\text{dom}(f) \cap \text{dom}(g)$
$\frac{f}{g}$	$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$	$\text{dom}(f) \cap \{x \in \text{dom}(g) | g(x) \neq 0\}$

Composite and Inverse Functions

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

• Inverse: $f^{-1}(y) = x$ if $f(x) = y$ and $f$ is one-to-one

• Horizontal Line Test: $f$ is one-to-one if every horizontal line meets its graph at most once.

Example: $f(x) = 3x + 5$ has inverse $f^{-1}(x) = \frac{x-5}{3}$.

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: $\text{Degrees} \to \text{Radians}:$ multiply by $\frac{\pi}{180}$; $\text{Radians} \to \text{Degrees}:$ multiply by $\frac{180}{\pi}$

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

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

Trigonometric Functions

• Definitions (unit circle):

  • $\sin \theta = \frac{y}{r}$

  • $\cos \theta = \frac{x}{r}$

  • $\tan \theta = \frac{y}{x}$

  • $\csc \theta = \frac{r}{y}$

  • $\sec \theta = \frac{r}{x}$

  • $\cot \theta = \frac{x}{y}$

• Periods: $\sin, \cos, \sec, \csc$ have period $2\pi$; $\tan, \cot$ have period $\pi$.

Special Angles Table:

$\theta$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
$\sin \theta$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos \theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$
$\tan \theta$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined

Trigonometric Identities

• $\sin^2 \theta + \cos^2 \theta = 1$

• $1 + \tan^2 \theta = \sec^2 \theta$

• $1 + \cot^2 \theta = \csc^2 \theta$

• $\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$

• $\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$

• $\sin(2\theta) = 2 \sin \theta \cos \theta$

• $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$

Inverse Trigonometric Functions

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

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

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

Solving Trigonometric Equations

• General solution for $\sin \theta = x$: $\theta = \arcsin x + 2k\pi$ or $\theta = \pi - \arcsin x + 2k\pi$, $k \in Z$

• General solution for $\cos \theta = x$: $\theta = \arccos x + 2k\pi$ or $\theta = -\arccos x + 2k\pi$

• General solution for $\tan \theta = x$: $\theta = \arctan x + k\pi$

Polar Coordinates

Definition and Conversion

• Polar coordinates: $(r, \theta)$ where $r \geq 0$ is the distance from the origin, $\theta$ is the angle from the positive x-axis.

• Conversion to Cartesian: $x = r \cos \theta$, $y = r \sin \theta$

• Conversion to Polar: $r = \sqrt{x^2 + y^2}$, $\theta = \arctan \frac{y}{x}$ (quadrant check required)

Polar Equations and Graphs

• Circle: $r = c$

• Line through origin: $\theta = k$

• Spiral: $r = k\theta$

• Rose: $r = a \cos n\theta$ or $r = a \sin n\theta$

• Cardioid: $r = a + b \sin \theta$ or $r = a + b \cos \theta$

Symmetry tests:

• x-axis: Replace $\theta$ with $-\theta$

• y-axis: Replace $\theta$ with $\pi - \theta$

• Origin: Replace $\theta$ with $\pi + \theta$

Mathematical Induction

Principle of Mathematical Induction

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

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

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

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

Sigma Notation and Binomial Theorem

Sigma Notation

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

• Properties:

  • $\sum_{i=r}^n (a_i + b_i) = \sum_{i=r}^n a_i + \sum_{i=r}^n b_i$

  • $\sum_{i=r}^n k a_i = k \sum_{i=r}^n a_i$

Summation Formulas

• $\sum_{r=1}^n r = \frac{n(n+1)}{2}$

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

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

Factorials and Binomial Coefficients

• Factorial: $n! = n \cdot (n-1) \cdots 2 \cdot 1$, $0! = 1$

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

Binomial Theorem

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

• Pascals Triangle: The coefficients $\binom{n}{k}$ form Pascal's triangle.

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

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:

• Parabola: One variable squared, one linear

• Ellipse: Both variables squared, same sign

• Hyperbola: Both variables squared, opposite signs

Transformations: Completing the square and rotation of axes can reduce any non-degenerate quadratic to canonical form.

Appendix: Basic Mathematical Notions

• Statement: An expression that is either true or false.

• Theorem: A true statement proved from axioms and definitions.

• Axiom: A statement assumed to be true without proof.

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

• Equivalence: "p if and only if q" ($p \Leftrightarrow q$)

• Contrapositive: The contrapositive of $p \Rightarrow q$ is $\neg q \Rightarrow \neg p$

• Proof by contradiction: Assume the negation and derive a contradiction.

Intervals:

Notation	Set Description	Type
$(a, b)$	$\{x \in R | a < x < b\}$	Open
$[a, b]$	$\{x \in R | a \leq x \leq b\}$	Closed
$[a, b)$	$\{x \in R | a \leq x < b\}$	Half-open
$(a, b]$	$\{x \in R | a < x \leq b\}$	Half-open
$(a, \infty)$	$\{x \in R | x > a\}$	Open
$[a, \infty)$	$\{x \in R | x \geq a\}$	Closed
$(-\infty, b)$	$\{x \in R | x < b\}$	Open
$(-\infty, b]$	$\{x \in R | x \leq b\}$	Closed

Additional info: These notes are structured to provide a comprehensive overview of foundational topics in college algebra and trigonometry, including definitions, properties, examples, and key formulas. For more advanced proofs and applications, consult the referenced textbook sections.