The Timpson Lyceum · Calculus 1 (2024)

Geometry, Trig, Sets, Relations, & Functions Review - Lecture 1 (Calculus 1)

Overview

This opening lecture is a supplementary review that lays the geometric, trigonometric, and set-theoretic foundations needed for calculus before the textbook is opened. It builds trigonometry from four core ideas — the radian, $\pi$, sine, and cosine — and shows how the unit circle encodes the coordinates of special angles. It then rebuilds the language of mathematics from the ground up: sets, relations, and functions, all defined precisely as collections of objects and ordered pairs. The goal is fluency: being able to fill in a unit circle quickly from memory and to reason about functions as sets of ordered pairs that get plotted as graphs.

Key Concepts

Angle sum of a triangle

• The three interior angles of any triangle sum to $180^\circ$.
• Proof idea: draw a line through one vertex parallel to the opposite side. The two outer angles equal the triangle's two base angles (alternate/corresponding angles where a transversal crosses parallel lines), and together with the top angle they form a straight angle of $180^\circ$.

Pythagorean theorem

• For a right triangle with legs $a$, $b$ and hypotenuse $c$: $a^2 + b^2 = c^2$.
• Geometric proof: arrange four copies of the triangle inside a square of side $a+b$. The outer area $(a+b)^2$ equals four triangle areas plus the inner square: $(a+b)^2 = 4\cdot\tfrac{1}{2}ab + c^2$. Expanding and cancelling the $2ab$ terms leaves $a^2 + b^2 = c^2$.

The four foundations of trig

• $\pi$ is defined as the ratio of circumference to diameter, $\pi = \tfrac{C}{D}$. The value $3.14159\ldots$ is just a numeric approximation; the ratio is the same for every circle. Rearranging gives $C = \pi D = 2\pi r$.
• Radian is the angle subtended by walking one radius along the circle. Walking all the way around ($C = 2\pi r$) means walking $2\pi$ radii, so a full circle is $2\pi$ radians. Radians make conversion between arc length and angle trivial: arc length $= r\theta$, and angle $= \tfrac{\text{arc length}}{r}$.
• Cosine of an angle is the $x$-coordinate of the corresponding point on the unit circle: $\cos\theta = \tfrac{x}{r}$.
• Sine of an angle is the $y$-coordinate on the unit circle: $\sin\theta = \tfrac{y}{r}$.

Degree–radian conversion

• A degree is the arbitrary unit of $\tfrac{1}{360}$ of a circle; a radian is tied to the geometry of the circle itself.
• Convert by finding what fraction of the circle the angle is and multiplying by $2\pi$: e.g. $60^\circ = \tfrac{1}{6}(2\pi) = \tfrac{\pi}{3}$, $90^\circ = \tfrac{\pi}{2}$, $180^\circ = \pi$, $270^\circ = \tfrac{3\pi}{2}$.

Scaling beyond the unit circle

• On a circle of radius $r$ (equivalently, a right triangle with hypotenuse $r$): $x = r\cos\theta$ and $y = r\sin\theta$. The unit circle is the special case $r = 1$.
• Solving for the function values gives $\cos\theta = \tfrac{x}{r}$ (base over hypotenuse) and $\sin\theta = \tfrac{y}{r}$ (height over hypotenuse). The circle is just a crutch — what matters is the triangle.

The six trig functions (all derived from sine and cosine)

• $\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{y}{x}$ — the rise over run, i.e. the slope of the line at angle $\theta$.
• $\sec\theta = \dfrac{1}{\cos\theta} = \dfrac{r}{x}$
• $\csc\theta = \dfrac{1}{\sin\theta} = \dfrac{r}{y}$
• $\cot\theta = \dfrac{1}{\tan\theta} = \dfrac{\cos\theta}{\sin\theta} = \dfrac{x}{y}$
• The six functions capture the six possible ratios among the three sides of a triangle.

Special-angle values (memorize the unit circle)

• $\sin\tfrac{\pi}{4} = \cos\tfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$ (from the isosceles right triangle).
• $\cos\tfrac{\pi}{3} = \tfrac{1}{2}$, $\sin\tfrac{\pi}{3} = \dfrac{\sqrt{3}}{2}$; and $\cos\tfrac{\pi}{6} = \dfrac{\sqrt{3}}{2}$, $\sin\tfrac{\pi}{6} = \tfrac{1}{2}$ (from the $30$–$60$–$90$ triangle, derived by reflecting into an equilateral triangle).
• Reflecting the reference triangle across the axes gives the values in all four quadrants (with appropriate signs).

Sets

• A set is an unordered collection of unique objects, written with braces: $\{1,2,3\}$. Order doesn't matter ($\{1,2,3\}=\{3,2,1\}$) and duplicates don't matter ($\{1,1,2,3,3\}=\{1,2,3\}$, three elements).
• Common sets: $\varnothing$ (empty set), $\mathbb{N}$ (naturals, including $0$), $\mathbb{Z}$ (integers), $\mathbb{Z}^+$ (positive integers), $\mathbb{Q}$ (rationals — fractions of integers, nonzero denominator), $\mathbb{R}$ (reals, e.g. $\sqrt{2}$ and $\pi$).
• Set-builder notation: $\{\, x \mid \text{condition} \,\}$. Example, the evens: $\{\, n \mid n \in \mathbb{Z} \text{ and } 2 \mid n \,\}$. The squares: $\{\, y \mid y = x^2,\ x \in \mathbb{Z} \,\}$.

Set relations (evaluate to true/false)

• Membership $x \in A$: $x$ is an element of $A$. Note $1 \in \{1,2,3\}$ but $\{1\} \notin \{1,2,3\}$.
• Subset $A \subseteq B$: every element of $A$ is also in $B$. Every set is a subset of itself, and $\varnothing \subseteq A$ for any $A$ (vacuously true).
• Equality $A = B$ iff $A \subseteq B$ and $B \subseteq A$.
• Proper subset $A \subsetneq B$: $A \subseteq B$ and $A \neq B$.

Set operations (take sets, return sets)

• Intersection $A \cap B = \{\, x \mid x \in A \text{ and } x \in B \,\}$ — elements in both.
• Union $A \cup B = \{\, x \mid x \in A \text{ or } x \in B \,\}$ — elements in at least one.
• Difference $A \setminus B = \{\, x \mid x \in A \text{ and } x \notin B \,\}$ — not commutative: $A \setminus B \neq B \setminus A$.
• Cross product $A \times B = \{\, (x,y) \mid x \in A,\ y \in B \,\}$ — all ordered pairs with first from $A$, second from $B$. $\cap$ and $\cup$ are commutative; $\setminus$ and $\times$ are not.

Lists, relations, and functions

• A list uses parentheses; order and duplicates matter, so $(1,2)\neq(2,1)$. A two-element list is an ordered pair. A list is never equal to a set.
• A relation is a set of ordered pairs. We write $a \,R\, b$ to mean $(a,b) \in R$; e.g. "$<$" is the infinite relation $\{(0,1),(0,2),(1,2),\dots\}$, so "$0 < 3$" just means $(0,3)$ is in that relation.
• A function is a relation in which each input maps to exactly one output: if $(a,b)\in f$ and $(a,c)\in f$, then $b=c$. We write $f(a)=b$ for $(a,b)\in f$. Example: $f = \{\,(x,y) \mid y = x^2\,\}$.

Graphs, domain, range, intervals

• To graph a function, interpret each ordered pair $(x,y)$ as the coordinates of a point and plot it; $f(x)=x^2$ plots as a parabola.
• The domain is the set of all first (input) values; the range is the set of all second (output) values. For $f(x)=x^2$ the domain is $\mathbb{R}$ and the range is $[0,\infty)$ — a square can't be negative.
• Interval notation: $(a,b)=\{\,x \mid a<x<b\,\}$ excludes endpoints; brackets $[a,b]$ include them; $\infty$ always takes a parenthesis, e.g. $[0,\infty)$.

Worked Example

Solve the triangle with hypotenuse $r = 8$ and base angle $\theta = \tfrac{\pi}{6}$, finding the horizontal leg $x$ and vertical leg $y$. Use the scaled unit-circle relations $x = r\cos\theta$ and $y = r\sin\theta$, with $\cos\tfrac{\pi}{6}=\tfrac{\sqrt{3}}{2}$ and $\sin\tfrac{\pi}{6}=\tfrac{1}{2}$.

So the triangle has horizontal leg $4\sqrt{3}$ and height $4$.

Summary

• Geometry foundations: the interior angles of any triangle sum to $180^\circ$, and the Pythagorean theorem $a^2+b^2=c^2$ can be proven by area-rearrangement of four right triangles.
• Trigonometry rests on four definitions — $\pi = C/D$, the radian (angle per one radius, so a full circle is $2\pi$), and $\cos\theta$ / $\sin\theta$ as the $x$- and $y$-coordinates on the unit circle; the other four functions are just ratios built from sine and cosine.
• Convert degrees to radians by taking the angle's fraction of the circle times $2\pi$; for any radius use $x=r\cos\theta$, $y=r\sin\theta$. Memorizing the unit circle (the $\tfrac{\sqrt{2}}{2}$ and $\tfrac{\sqrt{3}}{2},\tfrac{1}{2}$ values) makes special-angle work automatic.
• A set is an unordered, duplicate-free collection; relations ($\in$, $\subseteq$, $=$, $\subsetneq$) return true/false while operations ($\cap$, $\cup$, $\setminus$, $\times$) return new sets.
• Building up from sets: a relation is a set of ordered pairs, and a function is a relation that sends each input to exactly one output — written $f(x)$ and often given in set-builder form like $\{(x,y)\mid y=x^2\}$.
• Graphing means plotting a function's ordered pairs as points; the domain is the set of inputs and the range the set of outputs, both often expressed in interval notation where parentheses exclude endpoints and brackets include them.