12 KiB
| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| Trigonometry | Obsidian::STEM | 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.
%%ANKI Basic Trigonometry was originally the study of what geometric shape? Back: Triangles. Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
Functions
%%ANKI
Basic
What are the two most fundamental trigonometric functions?
Back: \sin and \cos.
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
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
Consider the following arc with length t on the unit circle. With maximum specificity, what is its terminal point?
!
Back: \langle \cos(t), \sin(t) \rangle
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%%
Sine
If the real number t is the directed length of an arc (either positive or negative) measured on the 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.
%%ANKI
Cloze
The {sine} of t is denoted as {\sin(t)}.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
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.
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.
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.
END%%
%%ANKI
Basic
Consider the following arc with length t on the unit circle. What is the terminal point's y-coordinate?
!
Back: \sin(t)
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
What does \sin(2\pi) evaluate to?
Back: 0
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
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(-\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
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%%
%%ANKI
Basic
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)?
Back: \sin(t) > 0
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
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)?
Back: \sin(t) > 0
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
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)?
Back: \sin(t) < 0
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
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)?
Back: \sin(t) < 0
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
Let t be the length of an arc with initial point \langle 1, 0 \rangle. When is \sin(t) > 0?
Back: When the terminal point of the arc is in quadrant I or II.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
Let t be the length of an arc with initial point \langle 1, 0 \rangle. When is \sin(t) < 0?
Back: When the terminal point of the arc is in quadrant III or IV.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
Let \frac{\pi}{2} < t < \pi. What is the sign of \sin(t)?
Back: Positive.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
Let \frac{3\pi}{2} < t < 2\pi. What is the sign of \sin(t)?
Back: Negative.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Cloze
For any real number t, {\sin^2(t)} is alternatively denoted as {(\sin(t))^2}.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
Cosine
If the real number t is the directed length of an arc (either positive or negative) measured on the 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.
%%ANKI
Cloze
The {cosine} of t is denoted as {\cos(t)}.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
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.
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.
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.
END%%
%%ANKI
Basic
Consider the following arc with length t on the unit circle. What is the terminal point's x-coordinate?
!
Back: \cos(t)
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
What does \cos(0) evaluate to?
Back: 1
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
What does \cos\left(\frac{\pi}{2}\right) evaluate to?
Back: 0
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
What does \cos\left(-\frac{\pi}{2}\right) evaluate to?
Back: 0
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
What does \cos\left(\pi\right) evaluate to?
Back: -1
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
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)?
Back: \cos(t) > 0
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
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)?
Back: \cos(t) < 0
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
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)?
Back: \cos(t) < 0
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
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)?
Back: \cos(t) < 0
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
Let t be the length of an arc with initial point \langle 1, 0 \rangle. When is \cos(t) > 0?
Back: When the terminal point of the arc is in quadrant I or IV.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
Let t be the length of an arc with initial point \langle 1, 0 \rangle. When is \cos(t) < 0?
Back: When the terminal point of the arc is in quadrant II or III.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
Let \frac{\pi}{2} < t < \pi. What is the sign of \cos(t)?
Back: Negative.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
Let \frac{3\pi}{2} < t < 2\pi. What is the sign of \cos(t)?
Back: Positive.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Cloze
For any real number t, {\cos^2(t)} is alternatively denoted as {(\cos(t))^2}.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
Identities
Pythagorean Identity
For any real number t, \cos^2(t) + \sin^2(t) = 1.
%%ANKI Basic Which trigonometric identity is often considered the most important? Back: The Pythagorean Identity. Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
What does the Pytagorean Identity state?
Back: For any real number t, \cos^2(t) + \sin^2(t) = 1.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI Basic What shape is used to derive the Pythagorean Identity? Back: The unit circle. Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
%%ANKI
Basic
How is the Pythagorean Identity derived?
Back: By applying the Pythagorean Theorem on point \langle \cos(t), \sin(t) \rangle on the unit circle.
Reference: Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
END%%
Bibliography
- Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.