Determine the coordinates of a point on the unit circle for a given angle $ heta$.

To determine the coordinates of a point on the unit circle for a given angle $\theta$, we use the fact that the unit circle has a radius of 1 and the coordinates can be expressed as $(\cos(\theta), \sin(\theta))$.

Let’s find the coordinates for $\theta = \frac{\pi}{4}$.

The cosine and sine of $\frac{\pi}{4}$ are as follows:

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

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

Thus, the coordinates of the point are:

$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$

To find the coordinates of a point on the unit circle for a given angle $ heta$, we use the unit circle definition where the coordinates are $(cos( heta), sin( heta))$.

Let’s calculate the coordinates for $ heta = frac{2pi}{3}$.

The cosine and sine of $frac{2pi}{3}$ are:

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

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

Therefore, the coordinates of the point are:

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

To determine the coordinates of a point on the unit circle for angle $ heta$, use $(cos( heta), sin( heta))$.

For $ heta = pi$,

$cos(pi) = -1$

$sin(pi) = 0$

The coordinates are:

$(-1, 0)$

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