Determine the coordinates of a point in the first quadrant of the unit circle given its angle $ heta $

To determine the coordinates of a point in the first quadrant on the unit circle given its angle $ \theta $, we use the trigonometric identities for sine and cosine:

$ x = \cos(\theta) $

$ y = \sin(\theta) $

For example, if $ \theta = \frac{\pi}{4} $:

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

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

So the coordinates are $ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $.

To find the coordinates of a point in the first quadrant on the unit circle given its angle $ heta $, use:

$ x = cos( heta), quad y = sin( heta) $

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

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

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

The coordinates are $ left( frac{1}{2}, frac{sqrt{3}}{2}
ight) $.

Given an angle $ heta $ in the first quadrant, the coordinates on the unit circle are:

$ x = cos( heta), quad y = sin( heta) $

For $ heta = frac{pi}{6} $:

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

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

So the coordinates are $ left( frac{sqrt{3}}{2}, frac{1}{2}
ight) $.

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