91Ó°ÊÓ

We are given the point \((-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\). Since it's on the unit circle, we know that its coordinates can be represented as \(( \cos\theta, \sin\theta)\) for some angle \(\theta\). We have the system of equations: \[ \begin{cases} \cos\theta = -\frac{\sqrt{2}}{2}, \\ \sin\theta = -\frac{\sqrt{2}}{2}. \end{cases} \] The angle \(\theta\) that satisfies this system is \(\theta = \frac{3\pi}{4}\) or \(135^{\circ}\).

We now have two angles on the unit circle, \(135^{\circ}\) and \(125^{\circ}\). To find the angles between these points, we calculate the absolute difference and its supplement. So we have: \[ \alpha_1 = |135^{\circ} - 125^{\circ}| = 10^{\circ} \] and \[ \alpha_2 = 360^{\circ} - \alpha_1 = 360^{\circ} - 10^{\circ} = 350^{\circ}. \] These are the angles in degrees, but we need to convert them to radians for the next step. To convert degrees to radians, we use the following formula: \[ \text{Radians} = \frac{\text{Degrees}}{180} \times \pi \] Thus, we have: \[ \alpha_1 = 10^{\circ} \times \frac{\pi}{180} = \frac{\pi}{18}\text{ radians} \] and \[ \alpha_2 = 350^{\circ} \times \frac{\pi}{180} = \frac{35\pi}{18}\text{ radians}. \]

Finally, we can calculate the lengths of the circular arcs using the angles we found in the previous step. Since we are dealing with a unit circle, the length of an arc is equal to the angle in radians: \[ \text{Arc length} = \text{Angle in radians} \times \text{radius} \] Since the radius of the unit circle is 1, the arc lengths will be equal to the angles in radians. Therefore, we have: \[ \text{Arc length}_1 = \alpha_1 = \frac{\pi}{18} \] and \[ \text{Arc length}_2 = \alpha_2 = \frac{35\pi}{18}. \] So the lengths of the two circular arcs connecting the given points on the unit circle are \(\frac{\pi}{18}\) and \(\frac{35\pi}{18}\).