For a unit circle centered at the origin, consider a point $P$ on the circle described by the angle $ heta$ measured from the positive x-axis in the counterclockwise direction. Determine the coordinates of the point $Q$, which is the reflection of $P$ ac
Answer 1
Matthew Carter
The coordinates of the point $P$ on the unit circle can be expressed as $P(\cos\theta, \sin\theta)$. To find the coordinates of the reflection of $P$ across the line $y=x$, we interchange the x and y coordinates of $P$. Therefore, the coordinates of $Q$ are $Q(\sin\theta, \cos\theta)$.
Let’s verify:
Given $P(\cos\theta, \sin\theta)$, reflecting across $y=x$ gives $Q(\sin\theta, \cos\theta)$.
Thus, the coordinates of $Q$ are $Q(\sin\theta, \cos\theta)$.
Answer 2
Emily Hall
The coordinates of $P$ on the unit circle are $P(cos heta, sin heta)$. To find the reflection of $P$ across the line $y=x$, we swap the x and y coordinates.
Hence, the coordinates of $Q$ are $Q(sin heta, cos heta)$.
Verification:
Starting with $P(cos heta, sin heta)$ and reflecting across $y=x$, we get $Q(sin heta, cos heta)$.
Therefore, $Q$ is located at $Q(sin heta, cos heta)$.
Answer 3
Daniel Carter
For $P(cos heta, sin heta)$ on a unit circle, the reflection across $y=x$ swaps the coordinates.
Therefore, $Q$ is at $Q(sin heta, cos heta)$.
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