429 lines
12 KiB
Markdown
429 lines
12 KiB
Markdown
---
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title: Trigonometry
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TARGET DECK: Obsidian::STEM
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FILE TAGS: trigonometry
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tags:
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- trigonometry
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---
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## Overview
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Trigonometry was originally derived from a Greek word meaning "triangle measuring". It has since been generalized to refer to the study of periodicity.
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%%ANKI
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Basic
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Trigonometry was originally the study of what geometric shape?
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Back: Triangles.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737167693405-->
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END%%
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## Functions
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%%ANKI
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Basic
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What are the two most fundamental trigonometric functions?
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Back: $\sin$ and $\cos$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513249-->
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END%%
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%%ANKI
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Cloze
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The {1:$x$}-coordinate is to {2:$\cos$} whereas the {2:$y$}-coordinate is to {1:$\sin$}.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513254-->
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END%%
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%%ANKI
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Basic
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Suppose an arc on the unit circle has terminal point $\langle \cos(t), \sin(t) \rangle$. What was its initial point?
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Back: $\langle 1, 0 \rangle$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513255-->
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END%%
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%%ANKI
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Basic
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Consider the following arc with length $t$ on the unit circle. With maximum specificity, what is its terminal point?
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![[example-arc.png]]
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Back: $\langle \cos(t), \sin(t) \rangle$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513262-->
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END%%
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%%ANKI
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Basic
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Why are $\sin$ and $\cos$ called circular functions?
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Back: Their values are determined by coordinates of points on the unit circle.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971452-->
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END%%
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### Sine
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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 **sine** of $t$, denoted $\sin(t)$ is defined to be $$\sin(t) = y.$$
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%%ANKI
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Cloze
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The {sine} of $t$ is denoted as {$\sin(t)$}.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513250-->
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END%%
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%%ANKI
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Basic
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Map $[0, t]$ to the unit circle. Geometrically, what does $\sin(t)$ correspond to?
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Back: The $y$-coordinate of the arc's terminal point.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513253-->
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END%%
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%%ANKI
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Basic
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What geometric aspect of the unit circle corresponds to the input of sine?
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Back: Arc length.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513259-->
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END%%
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%%ANKI
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Basic
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What geometric aspect of the unit circle corresponds to the output of the sine function?
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Back: The $y$-coordinate of an arc's terminal point.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513261-->
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END%%
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%%ANKI
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Basic
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Consider the following arc with length $t$ on the unit circle. What is the terminal point's $y$-coordinate?
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![[example-arc.png]]
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Back: $\sin(t)$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513256-->
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END%%
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%%ANKI
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Basic
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What does $\sin(2\pi)$ evaluate to?
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Back: $0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971447-->
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END%%
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%%ANKI
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Basic
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What does $\sin\left(\frac{\pi}{2}\right)$ evaluate to?
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Back: $1$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971449-->
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END%%
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%%ANKI
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Basic
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What does $\sin\left(-\frac{\pi}{2}\right)$ evaluate to?
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Back: $-1$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971450-->
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END%%
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%%ANKI
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Basic
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What does $\sin\left(\pi\right)$ evaluate to?
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Back: $0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971451-->
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END%%
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%%ANKI
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Basic
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What is the domain of $\sin$?
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Back: $\mathbb{R}$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971455-->
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END%%
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%%ANKI
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Basic
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What is the range of $\sin$?
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Back: $[-1, 1]$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971456-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$ and terminal point in quadrant I. What is the sign of $\sin(t)$?
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Back: $\sin(t) > 0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715565-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$ and terminal point in quadrant II. What is the sign of $\sin(t)$?
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Back: $\sin(t) > 0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715568-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$ and terminal point in quadrant III. What is the sign of $\sin(t)$?
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Back: $\sin(t) < 0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715575-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$ and terminal point in quadrant IV. What is the sign of $\sin(t)$?
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Back: $\sin(t) < 0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715578-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$. When is $\sin(t) > 0$?
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Back: When the terminal point of the arc is in quadrant I or II.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715588-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$. When is $\sin(t) < 0$?
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Back: When the terminal point of the arc is in quadrant III or IV.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715591-->
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END%%
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%%ANKI
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Basic
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Let $\frac{\pi}{2} < t < \pi$. What is the sign of $\sin(t)$?
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Back: Positive.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737417172718-->
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END%%
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%%ANKI
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Basic
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Let $\frac{3\pi}{2} < t < 2\pi$. What is the sign of $\sin(t)$?
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Back: Negative.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737417172722-->
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END%%
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%%ANKI
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Cloze
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For any real number $t$, {$\sin^2(t)$} is alternatively denoted as {$(\sin(t))^2$}.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737418825316-->
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END%%
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### Cosine
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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 $\cos(t)$, is defined to be $$\cos(t) = x.$$
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%%ANKI
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Cloze
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The {cosine} of $t$ is denoted as {$\cos(t)$}.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513251-->
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END%%
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%%ANKI
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Basic
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Map $[0, t]$ to the unit circle. Geometrically, what does $\cos(t)$ correspond to?
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Back: The $x$-coordinate of the arc's terminal point.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513252-->
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END%%
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%%ANKI
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Basic
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What geometric aspect of the unit circle corresponds to the input of the cosine function?
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Back: Arc length.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513257-->
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END%%
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%%ANKI
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Basic
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What geometric aspect of the unit circle corresponds to the output of the cosine function?
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Back: The $x$-coordinate of an arc's terminal point.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513258-->
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END%%
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%%ANKI
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Basic
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Consider the following arc with length $t$ on the unit circle. What is the terminal point's $x$-coordinate?
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![[example-arc.png]]
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Back: $\cos(t)$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349513260-->
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END%%
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%%ANKI
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Basic
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What does $\cos(0)$ evaluate to?
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Back: $1$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971441-->
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END%%
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%%ANKI
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Basic
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What does $\cos\left(\frac{\pi}{2}\right)$ evaluate to?
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Back: $0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971443-->
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END%%
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%%ANKI
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Basic
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What does $\cos\left(-\frac{\pi}{2}\right)$ evaluate to?
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Back: $0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971445-->
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END%%
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%%ANKI
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Basic
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What does $\cos\left(\pi\right)$ evaluate to?
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Back: $-1$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971446-->
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END%%
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%%ANKI
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Basic
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What is the domain of $\cos$?
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Back: $\mathbb{R}$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971453-->
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END%%
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%%ANKI
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Basic
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What is the range of $\cos$?
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Back: $[-1, 1]$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737349971454-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$ and terminal point in quadrant I. What is the sign of $\cos(t)$?
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Back: $\cos(t) > 0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715551-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$ and terminal point in quadrant II. What is the sign of $\cos(t)$?
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Back: $\cos(t) < 0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715557-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$ and terminal point in quadrant III. What is the sign of $\cos(t)$?
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Back: $\cos(t) < 0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715560-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$ and terminal point in quadrant IV. What is the sign of $\cos(t)$?
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Back: $\cos(t) < 0$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715562-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$. When is $\cos(t) > 0$?
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Back: When the terminal point of the arc is in quadrant I or IV.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715582-->
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END%%
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%%ANKI
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Basic
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Let $t$ be the length of an arc with initial point $\langle 1, 0 \rangle$. When is $\cos(t) < 0$?
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Back: When the terminal point of the arc is in quadrant II or III.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737416715585-->
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END%%
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%%ANKI
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Basic
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Let $\frac{\pi}{2} < t < \pi$. What is the sign of $\cos(t)$?
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Back: Negative.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737417172726-->
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END%%
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%%ANKI
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Basic
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Let $\frac{3\pi}{2} < t < 2\pi$. What is the sign of $\cos(t)$?
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Back: Positive.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737417172730-->
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END%%
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%%ANKI
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Cloze
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For any real number $t$, {$\cos^2(t)$} is alternatively denoted as {$(\cos(t))^2$}.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737418825321-->
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END%%
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## Identities
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### Pythagorean Identity
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For any real number $t$, $$\cos^2(t) + \sin^2(t) = 1.$$
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%%ANKI
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Basic
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Which trigonometric identity is often considered the most important?
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Back: The Pythagorean Identity.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737418825324-->
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END%%
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%%ANKI
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Basic
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What does the Pytagorean Identity state?
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Back: For any real number $t$, $\cos^2(t) + \sin^2(t) = 1$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737418825326-->
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END%%
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%%ANKI
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Basic
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What shape is used to derive the Pythagorean Identity?
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Back: The unit circle.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737418825329-->
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END%%
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%%ANKI
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Basic
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How is the Pythagorean Identity derived?
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Back: By applying the Pythagorean Theorem on point $\langle \cos(t), \sin(t) \rangle$ on the unit circle.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737418825332-->
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END%%
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## Bibliography
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* Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. |