215 lines
5.7 KiB
Markdown
215 lines
5.7 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 generalized to studying 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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## Unit Circle
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On the [[cartesian|Cartesian coordinate system]], the **unit circle** is the [[circle]] with center at the origin and radius $1$.
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%%ANKI
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Basic
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On the Cartesian coordinate system, what is the unit circle?
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Back: The circle with center at the origin and radius $1$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737167693410-->
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END%%
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%%ANKI
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Basic
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On the Cartesian coordinate system, where is the center of the unit circle located?
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Back: At $\langle 0, 0 \rangle$, i.e. the origin.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737167693413-->
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END%%
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%%ANKI
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Basic
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What is the radius of the unit circle?
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Back: $1$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737167693416-->
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END%%
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%%ANKI
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Basic
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What is the diameter of the unit circle?
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Back: $2$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737167693419-->
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END%%
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%%ANKI
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Basic
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What is the circumference of the unit circle?
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Back: $2\pi$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737167693428-->
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END%%
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%%ANKI
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Basic
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What is the area of the unit circle?
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Back: $\pi$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737167693435-->
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END%%
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%%ANKI
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Basic
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Which real numbers does the point $\langle 0, 0 \rangle$ on the unit circle map to?
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Back: N/A. This point is not on the circle itself.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737168795237-->
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END%%
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%%ANKI
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Basic
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Which real numbers does the point $\langle 1, 0 \rangle$ on the unit circle map to?
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Back: $2\pi k$ for all $k \in \mathbb{Z}$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737168795241-->
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END%%
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%%ANKI
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Basic
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Which point on the unit circle does number $2\pi$ map to?
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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: 1737168795265-->
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END%%
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%%ANKI
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Basic
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Which point on the unit circle does number $\frac{3\pi}{2}$ map to?
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Back: $\langle 0, -1 \rangle$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737168795244-->
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END%%
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%%ANKI
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Basic
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Which real numbers does the point $\langle 0, -1 \rangle$ on the unit circle map to?
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Back:$\frac{3\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737168795269-->
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END%%
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%%ANKI
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Basic
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Which real numbers does the point $\langle 0, 1 \rangle$ on the unit circle map to?
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Back: $\frac{\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737168795248-->
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END%%
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%%ANKI
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Basic
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Which point on the unit circle does number $\frac{\pi}{2}$ map to?
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Back: $\langle 0, 1 \rangle$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737168795261-->
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END%%
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%%ANKI
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Basic
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Which point on the unit circle does number $\pi$ map to?
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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: 1737168795252-->
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END%%
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%%ANKI
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Basic
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Which real numbers does the point $\langle -1, 0 \rangle$ on the unit circle map to?
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Back: $\pi + 2\pi k$ for all $k \in \mathbb{Z}$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737168795256-->
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END%%
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%%ANKI
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Basic
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Which real numbers correspond to the highlighted point on the unit circle?
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![[unit-circle-1-0.png]]
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Back: $2 \pi k$ for all $k \in \mathbb{Z}$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737169243685-->
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END%%
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%%ANKI
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Basic
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Which real numbers correspond to the highlighted point on the unit circle?
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![[unit-circle-0-1.png]]
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Back: $\frac{\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737169243690-->
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END%%
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%%ANKI
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Basic
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Which real numbers correspond to the highlighted point on the unit circle?
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![[unit-circle-n1-0.png]]
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Back: $\pi + 2\pi k$ for all $k \in \mathbb{Z}$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737169243692-->
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END%%
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%%ANKI
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Basic
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Which real numbers correspond to the highlighted point on the unit circle?
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![[unit-circle-0-n1.png]]
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Back: $\frac{3\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737169243695-->
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END%%
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%%ANKI
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Basic
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*Why* does point $\langle 1, 0 \rangle$ on the unit circle coincide with real number $2\pi$?
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Back: Because the circumference of the unit circle is $2\pi$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737169683142-->
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END%%
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%%ANKI
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Basic
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*Why* does point $\langle -1, 0 \rangle$ on the unit circle coincide with real number $\pi$?
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Back: Because half the circumference of the unit circle is $\pi$.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737169683151-->
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END%%
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%%ANKI
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Basic
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What is the "periodicity" of the unit circle?
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Back: $2 \pi$
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737230158153-->
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END%%
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%%ANKI
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Basic
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What property of the unit circle does its periodicity correspond to?
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Back: Its circumference.
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Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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<!--ID: 1737230158163-->
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END%%
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## Bibliography
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* Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. |