From 893334a81d4d62216d13feba3f34e7c8ca036777 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Sun, 19 Jan 2025 22:13:56 -0700 Subject: [PATCH] More basic trigonometry notes. --- .../plugins/obsidian-to-anki-plugin/data.json | 12 +- notes/_journal/2025-01-19.md | 3 +- notes/geometry/circle.md | 13 +- notes/trigonometry/images/example-arc.png | Bin 0 -> 12783 bytes notes/trigonometry/index.md | 230 +++++++------ notes/trigonometry/unit-circle.md | 316 ++++++++++++++++++ 6 files changed, 464 insertions(+), 110 deletions(-) create mode 100644 notes/trigonometry/images/example-arc.png create mode 100644 notes/trigonometry/unit-circle.md diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 3518b2d..c83d632 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -257,7 +257,8 @@ "function-general.png", "function-kernel.png", "triangular-gnomon.png", - "pascals-triangle.png" + "pascals-triangle.png", + "example-arc.png" ], "File Hashes": { "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8", @@ -1464,12 +1465,13 @@ "_journal/2025-01-16.md": "e3a21059205784a4e88bfe3b4deac7f7", "_journal/2025-01-17.md": "08a5f05bb572db9495bfc2b4feb8e0a9", "_journal/2025-01/2025-01-16.md": "e3a21059205784a4e88bfe3b4deac7f7", - "trigonometry/index.md": "6d07d5ba5e352f182a19b8a3d804321b", - "geometry/circle.md": "751a5aada6521281f1d8edb463572943", + "trigonometry/index.md": "06c82d1cf3789cc8cefd37e6b64fdc4a", + "geometry/circle.md": "9469e5512a6a92a962985eab99d00606", "_journal/2025-01-18.md": "8655fbf94aeec13efe9b6d2087c1f37e", "_journal/2025-01/2025-01-17.md": "08a5f05bb572db9495bfc2b4feb8e0a9", - "_journal/2025-01-19.md": "a37c6f534cf5e272619c5f813974afcf", - "_journal/2025-01/2025-01-18.md": "7a1655887093f37ffe86309d90459b3b" + "_journal/2025-01-19.md": "2acb8c3c37b9a44f06bac4903efed5ad", + "_journal/2025-01/2025-01-18.md": "7a1655887093f37ffe86309d90459b3b", + "trigonometry/unit-circle.md": "4a8fb4b5ac1bc2b645f18b0b8079897c" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2025-01-19.md b/notes/_journal/2025-01-19.md index ef1a9c3..596cd82 100644 --- a/notes/_journal/2025-01-19.md +++ b/notes/_journal/2025-01-19.md @@ -8,4 +8,5 @@ title: "2025-01-19" - [ ] Sheet Music (10 min.) - [ ] Korean (Read 1 Story) -* More notes on relocation entries. \ No newline at end of file +* More notes on relocation entries. +* More notes on basic trigonometry. \ No newline at end of file diff --git a/notes/geometry/circle.md b/notes/geometry/circle.md index 28e99d3..c4e9a8a 100644 --- a/notes/geometry/circle.md +++ b/notes/geometry/circle.md @@ -13,7 +13,7 @@ A **circle** is a shape consisting of all points in a plane at a given distance On the [[cartesian|Cartesian coordinate system]], the equation of a circle with radius $r$ and center $\langle h, k \rangle$ is $$(x - h)^2 + (y - k)^2 = r^2.$$ -Such a circle has circumference $2\pi r$ and [[area]] $\pi r^2$. +Such a circle has circumference $2\pi r$ and [[area]] $\pi r^2$. A continuous segment along the circumference of a circle is called an **arc**. %%ANKI Basic @@ -195,6 +195,15 @@ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia. END%% +%%ANKI +Basic +What term is used to describe a continuous portion of a circle? +Back: An arc. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + ## Bibliography -* “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). \ No newline at end of file +* “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). +* Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. \ No newline at end of file diff --git a/notes/trigonometry/images/example-arc.png b/notes/trigonometry/images/example-arc.png new file mode 100644 index 0000000000000000000000000000000000000000..a90ccb9b2969661d79b6eed9da0af5728936257d GIT binary patch literal 12783 zcmeHtXH-<#w(c%+PLgAR0aPf8oO2e*s6?eGikvAVDLDrLX@ZHYh=K}=k&FroDk3Td 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y-yVZzC>o?(@#OMQ4BF%VFOB;DvGJRW+hpaAYsY;v40Z3x@{uN!9`2Pb_dy75* literal 0 HcmV?d00001 diff --git a/notes/trigonometry/index.md b/notes/trigonometry/index.md index 41493ea..7f936d5 100644 --- a/notes/trigonometry/index.md +++ b/notes/trigonometry/index.md @@ -8,7 +8,9 @@ tags: ## Overview -Trigonometry was originally derived from a Greek word meaning "triangle measuring". It has generalized to studying periodicity. +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 @@ -18,196 +20,220 @@ Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. END%% -## Unit Circle - -On the [[cartesian|Cartesian coordinate system]], the **unit circle** is the [[circle]] with center at the origin and radius $1$. - %%ANKI Basic -On the Cartesian coordinate system, what is the unit circle? -Back: The circle with center at the origin and radius $1$. +What are the two most fundamental trigonometric functions? +Back: $\sin$ and $\cos$. Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - + +END%% + +%%ANKI +Cloze +The {sine} of $t$ is denoted as {$\sin(t)$}. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Cloze +The {cosine} of $t$ is denoted as {$\cos(t)$}. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + 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. +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. - + END%% %%ANKI Basic -What is the radius of the unit circle? -Back: $1$ +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. - + +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. + END%% %%ANKI Basic -What is the diameter of the unit circle? -Back: $2$ -Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - -END%% - -%%ANKI -Basic -What is the circumference of the unit circle? -Back: $2\pi$ -Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - -END%% - -%%ANKI -Basic -What is the area of the unit circle? -Back: $\pi$ -Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - -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. - -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. - -END%% - -%%ANKI -Basic -Which point on the unit circle does number $2\pi$ map to? +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. - + END%% %%ANKI Basic -Which point on the unit circle does number $\frac{3\pi}{2}$ map to? -Back: $\langle 0, -1 \rangle$ +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. - + 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 $k \in \mathbb{Z}$. +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. - + 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 $k \in \mathbb{Z}$. +What geometric aspect of the unit circle corresponds to the input of sine? +Back: Arc length. Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - + END%% %%ANKI Basic -Which point on the unit circle does number $\frac{\pi}{2}$ map to? -Back: $\langle 0, 1 \rangle$ +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. - + END%% %%ANKI Basic -Which point on the unit circle does number $\pi$ map to? -Back: $\langle -1, 0 \rangle$ +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. - + 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 $k \in \mathbb{Z}$. +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. - + 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 $k \in \mathbb{Z}$. +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. - + 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 $k \in \mathbb{Z}$. +What does $\cos(0)$ evaluate to? +Back: $1$ Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - + 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 $k \in \mathbb{Z}$. +What does $\cos\left(\frac{\pi}{2}\right)$ evaluate to? +Back: $0$ Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - + 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 $k \in \mathbb{Z}$. +What does $\cos\left(-\frac{\pi}{2}\right)$ evaluate to? +Back: $0$ Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - + 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 $2\pi$. +What does $\cos\left(\pi\right)$ evaluate to? +Back: $-1$ Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - + 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 $\pi$. +What does $\sin(2\pi)$ evaluate to? +Back: $0$ Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - + END%% %%ANKI Basic -What is the "periodicity" of the unit circle? -Back: $2 \pi$ +What does $\sin\left(\frac{\pi}{2}\right)$ evaluate to? +Back: $1$ Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - + END%% %%ANKI Basic -What property of the unit circle does its periodicity correspond to? -Back: Its circumference. +What does $\sin\left(-\frac{\pi}{2}\right)$ evaluate to? +Back: $-1$ Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. - + +END%% + +%%ANKI +Basic +What does $\sin\left(\pi\right)$ evaluate to? +Back: $0$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +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. + +END%% + +%%ANKI +Basic +What is the domain of $\cos$? +Back: $\mathbb{R}$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What is the range of $\cos$? +Back: $[-1, 1]$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What is the domain of $\sin$? +Back: $\mathbb{R}$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What is the range of $\sin$? +Back: $[-1, 1]$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + END%% ## Bibliography diff --git a/notes/trigonometry/unit-circle.md b/notes/trigonometry/unit-circle.md new file mode 100644 index 0000000..5717f23 --- /dev/null +++ b/notes/trigonometry/unit-circle.md @@ -0,0 +1,316 @@ +--- +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. + +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. + +END%% + +%%ANKI +Basic +What is the radius of the unit circle? +Back: $1$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What is the diameter of the unit circle? +Back: $2$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What is the circumference of the unit circle? +Back: $2\pi$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What is the area of the unit circle? +Back: $\pi$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +END%% + +%%ANKI +Basic +What is the "periodicity" of the unit circle? +Back: $2 \pi$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +END%% + +## Bibliography + +* Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. \ No newline at end of file