Given a point on the unit circle in Cartesian coordinates, find its other coordinate and the angle it makes with the positive x-axis.

To solve this problem, we start by using the unit circle equation:

$x^2 + y^2 = 1$

Given a point (x, y) = (\frac{1}{2}, y), we need to find y. Substitute x into the equation:

$\left(\frac{1}{2}\right)^2 + y^2 = 1$

$\frac{1}{4} + y^2 = 1$

$y^2 = 1 – \frac{1}{4}$

$y^2 = \frac{3}{4}$

$y = \pm \frac{\sqrt{3}}{2}$

So the points on the unit circle are (\frac{1}{2}, \frac{\sqrt{3}}{2}) and (\frac{1}{2}, -\frac{\sqrt{3}}{2}).

To find the angle with the positive x-axis:

$\cos \theta = \frac{1}{2}$

$\theta = \pm \frac{\pi}{3}$

The unit circle equation is:

$x^2 + y^2 = 1$

For the point (x, y) = (frac{sqrt{2}}{2}, y), substitute x into the equation:

$left(frac{sqrt{2}}{2}
ight)^2 + y^2 = 1$

$frac{2}{4} + y^2 = 1$

$frac{1}{2} + y^2 = 1$

$y^2 = frac{1}{2}$

$y = pm frac{sqrt{2}}{2}$

So the points on the unit circle are (frac{sqrt{2}}{2}, frac{sqrt{2}}{2}) and (frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}).

To find the angle with the positive x-axis:

$cos heta = frac{sqrt{2}}{2}$

$ heta = pm frac{pi}{4}$

Using the unit circle equation:

$x^2 + y^2 = 1$

If a point is (x, y) = (frac{sqrt{3}}{2}, y), substitute x:

$left(frac{sqrt{3}}{2}
ight)^2 + y^2 = 1$

$frac{3}{4} + y^2 = 1$

$y^2 = frac{1}{4}$

$y = pm frac{1}{2}$

So the points are (frac{sqrt{3}}{2}, frac{1}{2}) and (frac{sqrt{3}}{2}, -frac{1}{2}).

To find the angle with the positive x-axis:

$cos heta = frac{sqrt{3}}{2}$

$ heta = pm frac{pi}{6}$

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