Question

diagram shows coordinate system with origin o, upper - half - plane semicircle, point a with dashed horizontal line to y - axis (length 1) and dashed vertical line to x - axis (length √3), angle at o marked ?, and intersecting x - axis, y - axis.

Explanation:

Step1: Identify triangle type

The triangle formed is a right triangle with opposite side \(1\), adjacent side \(\sqrt{3}\), hypotenuse (radius) \(r\). First, find hypotenuse using Pythagoras: \(r = \sqrt{1^2 + (\sqrt{3})^2}\)
\(r = \sqrt{1 + 3} = \sqrt{4} = 2\)

Step2: Use trigonometric ratio

We can use \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\tan\theta=\frac{1}{\sqrt{3}}\)
We know that \(\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}\), so \(\theta = \frac{\pi}{6}\) (or \(30^\circ\))

Answer:

The angle (let's say \(\theta\)) is \(\frac{\pi}{6}\) (or \(30^\circ\))