The unit circle is the circle of radius one centered at the origin $\displaystyle (0,0)$ in the Cartesian coordinate system. $\displaystyle -\pi$ is equivalent to $\displaystyle -180^{\circ}$ which corresponds to the point $\displaystyle (-1,0)$ on the unit circle.

The unit circle is the circle of radius one centered at the origin $\displaystyle (0,0)$ in the Cartesian coordinate system. $\displaystyle \frac{5\pi}{2}$ radians is equivalent to $\displaystyle \frac{5\pi}{2}\times \frac{180^{\circ}}{\pi}=450^{\circ}$. This is a full circle $\displaystyle 360^{\circ}$ plus a quarter-turn $\displaystyle 90^{\circ}$ more. So, the angle $\displaystyle \frac{5\pi}{2}$ corresponds to the point $\displaystyle (0,1)$ on the unit circle.

The angle $\displaystyle \pi$ or $\displaystyle 180^{\circ}$ corresponds to the point $\displaystyle (-1,0)$.

The unit circle is the circle of radius one centered at the origin $\displaystyle (0,0)$ in the Cartesian coordinate system. $\displaystyle \frac{\pi}{2}$ is equivalent to $\displaystyle 90^{\circ}$ which corresponds to the point $\displaystyle (0,1)$ on the unit circle.

For a point to be on the unit circle, it has to have a radius of one.  Therefore, the sum of the squares of point's coordinates must also equal one.  

Let's try the point $\displaystyle (\frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2})$.

$\displaystyle (\frac{\sqrt{3}}{2})^{2} + (\frac{\sqrt{2}}{2})^{2} = \frac{5}{4} \neq 1$

Therefore, this point is not on the unit circle.

The coordinates of the point on the circle for each angle are $\displaystyle (\cos\theta,\sin\theta)$.

Since $\displaystyle \cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$ and $\displaystyle \sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, the point will be $\displaystyle (\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$.

By definition, the radius of the unit circle is 1. 

In order to find the point corresponds to the angle $\displaystyle \frac{\pi}{3}$ on the unit circle we can write:

$\displaystyle t=\frac{\pi}{3}\ or\ 60^{\circ}$

In the unit circle which has the radious of  $\displaystyle R=1$  we can write:

$\displaystyle cos\ t=\frac{x}{R}=\frac{x}{1}=x\Rightarrow x=cos(60^{\circ})=\frac{1}{2}$

$\displaystyle sin\ t=\frac{y}{R}=\frac{y}{1}=y\Rightarrow y=sin(60^{\circ})=\frac{\sqrt{3}}{2}$

So the point corresponds to the angle $\displaystyle \frac{\pi}{3}$ on the unit circle is $\displaystyle (\frac{1}{2},\frac{\sqrt{3}}{2})$

The unit circle is the circle of radius one centered at the origin $\displaystyle (0,0)$ in the Cartesian coordinate system. $\displaystyle 540^{\circ}$ is a full circle $\displaystyle 360^{\circ}$ plus a $\displaystyle 180^{\circ}$ more. So, the angle $\displaystyle 540^{\circ}$ corresponds to the point $\displaystyle (-1,0)$ on the unit circle.

The unit circle must have a radius of 1.

Use the circular area formula to find the area.

$\displaystyle A_{circle}= \pi r^2= \pi$