Find the angle $heta$ in radians for a point on the unit circle that satisfies given conditions

Given a point $P$ on the unit circle, where the coordinates of $P$ are $(\cos (\theta ),\sin (\theta ))$.

If the coordinates of $P$ are given as $(\frac{1}{2},\frac{\sqrt{3}}{2})$, we need to determine the angle $\theta$.

On the unit circle, these coordinates correspond to:

$\cos (\theta )=\frac{1}{2}\text{and}\sin (\theta )=\frac{\sqrt{3}}{2}$

From the unit circle, we know that:

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

Since the angle $\theta$ can also be in the second quadrant, we have:

$\theta =\frac{5\pi }{3}$

Given a point $P$ on the unit circle, with coordinates $(cos(heta),sin(heta))$.

If the coordinates of $P$ are $left(frac12,fracsqrt32ight)$, we need to find $heta$.

These coordinates correspond to:

$cos(heta)=frac12extandsin(heta)=fracsqrt32$

Thus:

$heta=fracpi3$

Or:

$heta=frac5pi3$

Given a point $P$ on the unit circle with coordinates $left(frac12,fracsqrt32ight)$, find $heta$.

These coordinates give:

$heta=fracpi3extorfrac5pi3$

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