Question

in the xy - plane above, o is the center of the circle, and the measure of ∠ros is equal to nπ radians. what is the value of n?

Explanation:

Step1: Find the distance from the origin to point \( R \)

Use the distance formula \( d=\sqrt{(x - 0)^2+(y - 0)^2} \). For \( R(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}) \), \( d=\sqrt{(-\frac{\sqrt{2}}{2})^2+(\frac{\sqrt{2}}{2})^2}=\sqrt{\frac{2}{4}+\frac{2}{4}} = 1 \). So the radius of the circle \( r = 1 \).

Step2: Use the cosine formula for the angle

\(\cos\angle ROS=\frac{x}{r}\) (since \( r = 1 \), \( x=-\frac{\sqrt{2}}{2} \)). \(\cos\angle ROS=-\frac{\sqrt{2}}{2}\).

Step3: Determine the angle in radians

We know that \(\cos\theta=-\frac{\sqrt{2}}{2}\) and \(\theta\in[0,2\pi]\). The angle \(\theta=\frac{3\pi}{4}\).

Answer:

\(\frac{3}{4}\)