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Lucas Brown
Given the point on the unit circle represented by $e^{i\theta}$, we begin by expressing this in terms of its Cartesian coordinates:
\[ e^{i\theta} = \cos(\theta) + i\sin(\theta) \]
When the unit circle is flipped over the real axis, the imaginary part changes sign. Thus, the new coordinates become:
\[ e^{i\theta} \rightarrow \cos(\theta) – i\sin(\theta) \]
Therefore, the new coordinates for the point on the flipped unit circle are:
\[ \boxed{\cos(\theta) – i\sin(\theta)} \]
Answer 2
Benjamin Clark
First, represent the point on the unit circle using Euler’s formula:
[ e^{i heta} = cos( heta) + isin( heta) ]
Flipping the unit circle over the real axis inverts the sign of the imaginary component. The new point is:
[ cos( heta) + i(-sin( heta)) ]
This simplifies to:
[ cos( heta) – isin( heta) ]
So, the coordinates of the new point after flipping are:
[ oxed{cos( heta) – isin( heta)} ]
Answer 3
Amelia Mitchell
Start by expressing $e^{i heta}$ in Cartesian form:
[ e^{i heta} = cos( heta) + isin( heta) ]
After flipping over the real axis, the new coordinates are:
[ oxed{cos( heta) – isin( heta)} ]
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