Given a unit circle centered at the origin, find the coordinates of a point $P$ on the circle such that the angle $ heta$ between the line segment $OP$ and the positive x-axis is an irrational multiple of $pi$.

To solve this problem, we need to find the coordinates of point $P$ on the unit circle given that the angle $\theta$ is an irrational multiple of $\pi$. Let’s denote this angle as $\theta = k\pi$ where $k$ is an irrational number.

Using the parametric equations of the unit circle, we have:

$x = \cos(\theta)$

$y = \sin(\theta)$

Since $\theta$ is an irrational multiple of $\pi$, we can choose $\theta = \sqrt{2}\pi$. Then, the coordinates $(x,y)$ of point $P$ are:

$x = \cos(\sqrt{2}\pi)$

$y = \sin(\sqrt{2}\pi)$

Thus, the coordinates of $P$ are:

$P = (\cos(\sqrt{2}\pi), \sin(\sqrt{2}\pi))$

To find the coordinates of point $P$ on the unit circle where the angle $ heta$ is an irrational multiple of $pi$, we use the unit circle parametric equations:

$x = cos( heta)$

$y = sin( heta)$

Given $ heta = kpi$ with $k$ as an irrational number, let $ heta = pi e$ (where $e$ is the Euler’s number, known to be irrational). Then:

$x = cos(pi e)$

$y = sin(pi e)$

Therefore, the coordinates are:

$P = (cos(pi e), sin(pi e))$

For a unit circle, the coordinates of a point $P$ with an angle $ heta$ (irrational multiple of $pi$) are:

$x = cos( heta)$

$y = sin( heta)$

Using $ heta = pi phi$ (where $phi$ is the golden ratio and irrational), we get:

$P = (cos(pi phi), sin(pi phi))$

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