Find the coordinates of a point on the unit circle that corresponds to a complex exponential representation $e^{i heta}$ given a specific angle $ heta$ in radians.

To find the coordinates of a point on the unit circle corresponding to $e^{i\theta}$ where $\theta = \frac{5\pi}{4}$, we use Euler’s formula:

$e^{i\theta} = \cos(\theta) + i\sin(\theta)$

Substituting $\theta = \frac{5\pi}{4}$:

$e^{i\frac{5\pi}{4}} = \cos\left(\frac{5\pi}{4}\right) + i\sin\left(\frac{5\pi}{4}\right)$

From the unit circle, we know:

$\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

$\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

Thus, the coordinates are:

$\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$

Consider the point on the unit circle corresponding to the complex number $e^{i heta}$ where $ heta = frac{7pi}{6}$. Using Euler’s formula:

$e^{i heta} = cos( heta) + isin( heta)$

Substituting $ heta = frac{7pi}{6}$:

$e^{ifrac{7pi}{6}} = cosleft(frac{7pi}{6}
ight) + isinleft(frac{7pi}{6}
ight)$

From trigonometric identities:

$cosleft(frac{7pi}{6}
ight) = -frac{sqrt{3}}{2}$

$sinleft(frac{7pi}{6}
ight) = -frac{1}{2}$

Thus, the coordinates of the point are:

$left(-frac{sqrt{3}}{2}, -frac{1}{2}
ight)$

To find the coordinates of a point on the unit circle for $ heta = frac{11pi}{6}$, we use:

$e^{i heta} = cos( heta) + isin( heta)$

Substitute $ heta = frac{11pi}{6}$:

$e^{ifrac{11pi}{6}} = cosleft(frac{11pi}{6}
ight) + isinleft(frac{11pi}{6}
ight)$

We know:

$cosleft(frac{11pi}{6}
ight) = frac{sqrt{3}}{2}$

$sinleft(frac{11pi}{6}
ight) = -frac{1}{2}$

Therefore, the coordinates are:

$left(frac{sqrt{3}}{2}, -frac{1}{2}
ight)$

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