$Find the coordinates of a point on the unit circle given an angle in radians$

Given an angle $\theta = \frac{\pi}{3}$ radians, find the coordinates of the point on the unit circle.

The unit circle is defined by the equation $x^2 + y^2 = 1$. For an angle $\theta$, the coordinates $(x, y)$ can be found using:

$x = \cos(\theta)$

$y = \sin(\theta)$

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

$x = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

$y = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$

So, the coordinates of the point are $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

Given an angle $ heta = frac{3pi}{4}$ radians, find the coordinates of the point on the unit circle.

The unit circle follows $x^2 + y^2 = 1$. For any angle $ heta$, coordinates $(x, y)$ are:

$x = cos( heta)$

$y = sin( heta)$

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

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

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

So, the coordinates of the point are $left(-frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight)$.

Given an angle $ heta = pi$ radians, find the coordinates of the point on the unit circle.

On the unit circle, coordinates $(x, y)$ for $ heta$ are:

$x = cos( heta)$

$y = sin( heta)$

Substituting $ heta = pi$:

$x = cos(pi) = -1$

$y = sin(pi) = 0$

So, the coordinates of the point are $(-1, 0)$.

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