Find the Cartesian coordinates of a point on the unit circle when given an angle $ heta$ and a trigonometric function value.

Given the angle $\theta = \frac{7\pi}{6}$ on the unit circle, find the Cartesian coordinates $ (x, y) $ for the corresponding point.

Since the unit circle has a radius of 1, we use the trigonometric identities for sine and cosine:

$ x = \cos(\theta) $

$ y = \sin(\theta) $

For $\theta = \frac{7\pi}{6}$:

$ x = \cos\left(\frac{7\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} $

$ y = \sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2} $

Therefore, the Cartesian coordinates are:

$ (x, y) = \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) $

Given the angle $ heta = frac{5pi}{4}$ on the unit circle, find the Cartesian coordinates $ (x, y) $ for the corresponding point.

Since the unit circle has a radius of 1, we use the trigonometric identities for sine and cosine:

$ x = cos( heta) $

$ y = sin( heta) $

For $ heta = frac{5pi}{4}$:

$ x = cosleft(frac{5pi}{4}
ight) = -cosleft(frac{pi}{4}
ight) = -frac{sqrt{2}}{2} $

$ y = sinleft(frac{5pi}{4}
ight) = -sinleft(frac{pi}{4}
ight) = -frac{sqrt{2}}{2} $

Therefore, the Cartesian coordinates are:

$ (x, y) = left(-frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}
ight) $

Given the angle $ heta = frac{4pi}{3}$ on the unit circle, find the Cartesian coordinates $ (x, y) $.

For $ heta = frac{4pi}{3}$:

$ x = cosleft(frac{4pi}{3}
ight) = -frac{1}{2} $

$ y = sinleft(frac{4pi}{3}
ight) = -frac{sqrt{3}}{2} $

Therefore:

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

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