Given a point $P$ on the unit circle such that its coordinates are $(cos( heta), sin( heta))$, find the coordinates of the point $Q$, which is the reflection of $P$ across the line $y = x$. Then, find the coordinates of the point $R$, which is the refle
Answer 1
Samuel Scott
To find the coordinates of the point $Q$, which is the reflection of $P$ across the line $y = x$, we switch the coordinates of $P$. Therefore, the coordinates of $Q$ are $(sin(\theta), cos(\theta))$.
Next, to find the coordinates of the point $R$, which is the reflection of $Q$ across the $x$-axis, we negate the y-coordinate of $Q$. Thus, the coordinates of $R$ are $(sin(\theta), -cos(\theta))$.
Summary:
Coordinates of $Q$: $(sin(\theta), cos(\theta))$
Coordinates of $R$: $(sin(\theta), -cos(\theta))$
Answer 2
William King
We start with point $P$ on the unit circle with coordinates $(cos( heta), sin( heta))$. When reflecting $P$ across the line $y = x$, the coordinates become reversed. Therefore, point $Q$ is at $(sin( heta), cos( heta))$.
To reflect $Q$ across the $x$-axis, we change the sign of the y-coordinate of $Q$. Therefore, point $R$ has coordinates $(sin( heta), -cos( heta))$.
In conclusion:
$Q$: $(sin( heta), cos( heta))$
$R$: $(sin( heta), -cos( heta))$
Answer 3
Isabella Walker
Given $P$ as $(cos( heta), sin( heta))$. Reflecting $P$ across $y = x$ gives $Q$ as $(sin( heta), cos( heta))$.
Reflecting $Q$ across the $x$-axis gives $R$ as $(sin( heta), -cos( heta))$.
$Q$: $(sin( heta), cos( heta))$
$R$: $(sin( heta), -cos( heta))$
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