Question

in the xy - plane above, o is the center of the circle, and the measure of \\( \angle aob \\) is \\( \frac{\pi}{a} \\) radians. what is the value of a?

Explanation:

Step1: Calculate the radius of the circle

The radius \(r\) of the circle can be found using the distance formula for point \(A(1,1)\) from the origin \(O(0,0)\). The distance formula is \(r = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Substituting \(x_1 = 0,y_1 = 0,x_2 = 1,y_2 = 1\), we get \(r=\sqrt{(1 - 0)^2+(1 - 0)^2}=\sqrt{1 + 1}=\sqrt{2}\).

Step2: Use the cosine function to find the \(x\) - coordinate of point \(B\)

We know that for a point \((x,y)\) on a circle of radius \(r\) centered at the origin, \(x=r\cos\theta\), where \(\theta\) is the angle made with the positive \(x\) - axis. Here, \(\theta=\frac{\pi}{4}\) radians and \(r = \sqrt{2}\).
Since \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), then \(x=\sqrt{2}\times\cos\frac{\pi}{4}\).
Substitute \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\) into the formula: \(x=\sqrt{2}\times\frac{\sqrt{2}}{2}\).
Simplify the expression: \(x=\frac{2}{2}=1\).

Answer:

\(1\)