Trigonometry was originally derived from a Greek word meaning "triangle measuring". It has since been generalized to refer to the study of periodicity.
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.
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### Sine
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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Cloze
The {sine} of $t$ is denoted as {$\sin(t)$}.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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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.
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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.
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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.
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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.
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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.
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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.
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Basic
Let $\frac{\pi}{2} <t< \pi$.Whatisthesignof$\sin(t)$?
Back: Positive.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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Basic
Let $\frac{3\pi}{2} <t<2\pi$.Whatisthesignof$\sin(t)$?
Back: Negative.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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Cloze
For any real number $t$, {$\sin^2(t)$} is alternatively denoted as {$(\sin(t))^2$}.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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### Cosine
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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Cloze
The {cosine} of $t$ is denoted as {$\cos(t)$}.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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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.
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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.
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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.
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Basic
Consider the following arc with length $t$ on the unit circle. What is the terminal point's $x$-coordinate?
