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.

\(\displaystyle A_{circle}= \pi r^2= \pi\)