---
title: Unit Circle
TARGET DECK: Obsidian::STEM
FILE TAGS: trigonometry
tags:
  - trigonometry
  - unit-circle
---

## Overview

On the [[cartesian|Cartesian coordinate system]], the **unit circle** is the [[circle]] with center at the origin and radius $1$.

Suppose [[intervals|closed interval]] $[a, b]$ is mapped to an arc on the unit circle. Then the point on the unit circle corresponding to $a$ is called the **initial point** of the arc. The point corresponding to $b$ is called the **terminal point** of the arc.

%%ANKI
Basic
On the Cartesian coordinate system, what is the unit circle?
Back: The circle with center at the origin and radius $1$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693410-->
END%%

%%ANKI
Basic
On the Cartesian coordinate system, where is the center of the unit circle located?
Back: At $\langle 0, 0 \rangle$, i.e. the origin.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693413-->
END%%

%%ANKI
Basic
What is the radius of the unit circle?
Back: $1$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693416-->
END%%

%%ANKI
Basic
What is the diameter of the unit circle?
Back: $2$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693419-->
END%%

%%ANKI
Basic
What is the circumference of the unit circle?
Back: $2\pi$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693428-->
END%%

%%ANKI
Basic
What is the area of the unit circle?
Back: $\pi$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693435-->
END%%

%%ANKI
Basic
Which real numbers does the point $\langle 0, 0 \rangle$ on the unit circle map to?
Back: N/A. This point is not on the circle itself.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795237-->
END%%

%%ANKI
Basic
Which real numbers does the point $\langle 1, 0 \rangle$ on the unit circle map to?
Back: $2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795241-->
END%%

%%ANKI
Basic
Which point on the unit circle does number $2\pi$ map to?
Back: $\langle 1, 0 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795265-->
END%%

%%ANKI
Basic
Which point on the unit circle does number $\frac{3\pi}{2}$ map to?
Back: $\langle 0, -1 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795244-->
END%%

%%ANKI
Basic
Which real numbers does the point $\langle 0, -1 \rangle$ on the unit circle map to?
Back:$\frac{3\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795269-->
END%%

%%ANKI
Basic
Which real numbers does the point $\langle 0, 1 \rangle$ on the unit circle map to?
Back: $\frac{\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795248-->
END%%

%%ANKI
Basic
Which point on the unit circle does number $\frac{\pi}{2}$ map to?
Back: $\langle 0, 1 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795261-->
END%%

%%ANKI
Basic
Which point on the unit circle does number $\pi$ map to?
Back: $\langle -1, 0 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795252-->
END%%

%%ANKI
Basic
Which real numbers does the point $\langle -1, 0 \rangle$ on the unit circle map to?
Back: $\pi + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795256-->
END%%

%%ANKI
Basic
Which real numbers correspond to the highlighted point on the unit circle?
![[unit-circle-1-0.png]]
Back: $2 \pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169243685-->
END%%

%%ANKI
Basic
Which real numbers correspond to the highlighted point on the unit circle?
![[unit-circle-0-1.png]]
Back: $\frac{\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169243690-->
END%%

%%ANKI
Basic
Which real numbers correspond to the highlighted point on the unit circle?
![[unit-circle-n1-0.png]]
Back: $\pi + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169243692-->
END%%

%%ANKI
Basic
Which real numbers correspond to the highlighted point on the unit circle?
![[unit-circle-0-n1.png]]
Back: $\frac{3\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169243695-->
END%%

%%ANKI
Basic
*Why* does point $\langle 1, 0 \rangle$ on the unit circle coincide with real number $2\pi$?
Back: Because the circumference of the unit circle is $2\pi$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169683142-->
END%%

%%ANKI
Basic
*Why* does point $\langle -1, 0 \rangle$ on the unit circle coincide with real number $\pi$?
Back: Because half the circumference of the unit circle is $\pi$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169683151-->
END%%

%%ANKI
Basic
What is the "periodicity" of the unit circle?
Back: $2 \pi$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737230158153-->
END%%

%%ANKI
Basic
What property of the unit circle does its periodicity correspond to?
Back: Its circumference.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737230158163-->
END%%

%%ANKI
Basic
What arc describes the portion of a unit circle found in the first quadrant?
Back: $\left(0, \frac{\pi}{2}\right)$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347029839-->
END%%

%%ANKI
Basic
What arc describes the portion of a unit circle found in the third quadrant?
Back: $\left(\pi, \frac{3\pi}{2}\right)$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347029841-->
END%%

%%ANKI
Basic
What arc describes the portion of a unit circle found in the fourth quadrant?
Back: $\left(\frac{3\pi}{2}, 2\pi\right)$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347029843-->
END%%

%%ANKI
Basic
What arc describes the portion of a unit circle found in the second quadrant?
Back: $\left(\frac{\pi}{2}, \pi\right)$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347029844-->
END%%

%%ANKI
Basic
Suppose interval $[a, b]$ is mapped to the unit circle. What is the point corresponding to $a$ called?
Back: The initial point.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347029845-->
END%%

%%ANKI
Basic
Suppose interval $[a, b]$ is mapped to the unit circle. What is the point corresponding to $b$ called?
Back: The terminal point.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347029847-->
END%%

%%ANKI
Basic
Map $\left[0, \frac{\pi}{2}\right]$ onto the unit circle. What is the initial point of this arc?
Back: $\langle 1, 0 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347029848-->
END%%

%%ANKI
Basic
Map $\left[0, \frac{\pi}{2}\right]$ onto the unit circle. What is the terminal point of this arc?
Back: $\langle 0, 1 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347029849-->
END%%

%%ANKI
Basic
Map $\left(0, \frac{\pi}{2}\right)$ onto the unit circle. What is the initial point of this arc?
Back: N/A. There is no initial point.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347029850-->
END%%

%%ANKI
Basic
Map $\left(0, \frac{\pi}{2}\right)$ onto the unit circle. What is the terminal point of this arc?
Back: N/A. There is no terminal point.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347029851-->
END%%

%%ANKI
Basic
Map $\left[-\pi, -\frac{\pi}{2}\right]$ onto the unit circle. What is the initial point of this arc?
Back: $\langle -1, 0 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513245-->
END%%

%%ANKI
Basic
Map $\left[-\pi, -\frac{\pi}{2}\right]$ onto the unit circle. What is the terminal point of this arc?
Back: $\langle 0, -1 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737349513248-->
END%%

%%ANKI
Cloze
Moving in the {positive} direction of the real number line corresponds to moving {counterclockwise} on the unit circle.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347197473-->
END%%

%%ANKI
Cloze
Moving in the {negative} direction of the real number line corresponds to moving {clockwise} on the unit circle.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737347197477-->
END%%

## Bibliography

* Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.