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.
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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.
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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.
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What is the radius of the unit circle?
Back: $1$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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What is the diameter of the unit circle?
Back: $2$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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What is the circumference of the unit circle?
Back: $2\pi$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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What is the area of the unit circle?
Back: $\pi$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Which point on the unit circle does number $\pi$ map to?
Back: $\langle -1, 0 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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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.
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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.
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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.
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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.
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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.
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*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.
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*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.
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What is the "periodicity" of the unit circle?
Back: $2 \pi$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
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What property of the unit circle does its periodicity correspond to?
Back: Its circumference.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
