notebook/notes/trigonometry/index.md

---
title: Trigonometry
TARGET DECK: Obsidian::STEM
FILE TAGS: trigonometry
tags:
  - trigonometry
---

## Overview

Trigonometry was originally derived from a Greek word meaning "triangle measuring". It has since been generalized to refer to the study of periodicity.

If the real number $t$ is the directed length of an arc (either positive or negative) measured on the [[unit-circle|unit circle]] $x^2 + y^2 = 1$ (with counterclockwise as the positive direction) with initial point $\langle 1, 0 \rangle$ and terminal point $\langle x, y \rangle$, then the **cosine** of $t$, denoted $\cos(t)$, and **sine** of $t$, denoted $\sin(t)$, are defined to be $$\cos(t) = x \quad\text{and}\quad \sin(t) = y.$$

%%ANKI
Basic
Trigonometry was originally the study of what geometric shape?
Back: Triangles.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693405-->
END%%

%%ANKI
Basic
What are the two most fundamental trigonometric functions?
Back: $\sin$ and $\cos$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513249-->
END%%

%%ANKI
Cloze
The {sine} of $t$ is denoted as {$\sin(t)$}.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513250-->
END%%

%%ANKI
Cloze
The {cosine} of $t$ is denoted as {$\cos(t)$}.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513251-->
END%%

%%ANKI
Basic
Map $[0, t]$ to the unit circle. Geometrically, what does $\cos(t)$ correspond to?
Back: The $x$-coordinate of the arc's terminal point.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513252-->
END%%

%%ANKI
Basic
Map $[0, t]$ to the unit circle. Geometrically, what does $\sin(t)$ correspond to?
Back: The $y$-coordinate of the arc's terminal point.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513253-->
END%%

%%ANKI
Cloze
The {1:$x$}-coordinate is to {2:$\cos$} whereas the {2:$y$}-coordinate is to {1:$\sin$}.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513254-->
END%%

%%ANKI
Basic
Suppose an arc on the unit circle has terminal point $\langle \cos(t), \sin(t) \rangle$. What was its initial point?
Back: $\langle 1, 0 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513255-->
END%%

%%ANKI
Basic
What geometric aspect of the unit circle corresponds to the input of the cosine function?
Back: Arc length.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513257-->
END%%

%%ANKI
Basic
What geometric aspect of the unit circle corresponds to the output of the cosine function?
Back: The $x$-coordinate of an arc's terminal point.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513258-->
END%%

%%ANKI
Basic
What geometric aspect of the unit circle corresponds to the input of sine?
Back: Arc length.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513259-->
END%%

%%ANKI
Basic
What geometric aspect of the unit circle corresponds to the output of the sine function?
Back: The $y$-coordinate of an arc's terminal point.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513261-->
END%%

%%ANKI
Basic
Consider the following arc with length $t$ on the unit circle. What is the terminal point's $x$-coordinate?
![[example-arc.png]]
Back: $\cos(t)$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513260-->
END%%

%%ANKI
Basic
Consider the following arc with length $t$ on the unit circle. What is the terminal point's $y$-coordinate?
![[example-arc.png]]
Back: $\sin(t)$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513256-->
END%%

%%ANKI
Basic
Consider the following arc with length $t$ on the unit circle. With maximum specificity, what is its terminal point?
![[example-arc.png]]
Back: $\langle \cos(t), \sin(t) \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513262-->
END%%

%%ANKI
Basic
What does $\cos(0)$ evaluate to?
Back: $1$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971441-->
END%%

%%ANKI
Basic
What does $\cos\left(\frac{\pi}{2}\right)$ evaluate to?
Back: $0$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971443-->
END%%

%%ANKI
Basic
What does $\cos\left(-\frac{\pi}{2}\right)$ evaluate to?
Back: $0$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971445-->
END%%

%%ANKI
Basic
What does $\cos\left(\pi\right)$ evaluate to?
Back: $-1$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971446-->
END%%

%%ANKI
Basic
What does $\sin(2\pi)$ evaluate to?
Back: $0$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971447-->
END%%

%%ANKI
Basic
What does $\sin\left(\frac{\pi}{2}\right)$ evaluate to?
Back: $1$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971449-->
END%%

%%ANKI
Basic
What does $\sin\left(-\frac{\pi}{2}\right)$ evaluate to?
Back: $-1$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971450-->
END%%

%%ANKI
Basic
What does $\sin\left(\pi\right)$ evaluate to?
Back: $0$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971451-->
END%%

%%ANKI
Basic
Why are $\sin$ and $\cos$ called circular functions?
Back: Their values are determined by coordinates of points on the unit circle.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971452-->
END%%

%%ANKI
Basic
What is the domain of $\cos$?
Back: $\mathbb{R}$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971453-->
END%%

%%ANKI
Basic
What is the range of $\cos$?
Back: $[-1, 1]$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971454-->
END%%

%%ANKI
Basic
What is the domain of $\sin$?
Back: $\mathbb{R}$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971455-->
END%%

%%ANKI
Basic
What is the range of $\sin$?
Back: $[-1, 1]$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349971456-->
END%%

## Bibliography

* Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.