Question

the diagram shows an angle, graphed in the xy-coordinate plane. segment pt is the initial side of the angle, and segment ps is the terminal side. segments pt and ps are radii of the unit circle centered at the origin p(0, 0). the x-coordinate of point s is $\frac{-sqrt{3}}{2}$. what is the measure of $\theta$ to the nearest hundredth of a radian?
options: 1.05, 2.62, 3.67, 2.09

Explanation:

Step1: Recall Unit Circle Cosine

On the unit circle, the x - coordinate of a point \((x,y)\) on the unit circle centered at the origin is given by \(x = \cos\theta\), where \(\theta\) is the angle formed with the positive x - axis (initial side PT here). We know that the x - coordinate of point S is \(x=-\frac{\sqrt{3}}{2}\), so \(\cos\theta=-\frac{\sqrt{3}}{2}\).

Step2: Determine the Quadrant

From the diagram, the angle \(\theta\) is in the second quadrant (since the terminal side PS is in the second quadrant, between the positive y - axis and negative x - axis). We know that \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\), and in the second quadrant, \(\cos\theta =-\cos(\pi - \theta)\). So we are looking for an angle \(\theta\) in the second quadrant where \(\cos\theta=-\frac{\sqrt{3}}{2}\). The reference angle \(\alpha\) such that \(\cos\alpha=\frac{\sqrt{3}}{2}\) is \(\alpha = \frac{\pi}{6}\approx0.5236\) radians. In the second quadrant, \(\theta=\pi-\alpha\).

Step3: Calculate the Angle

\(\theta=\pi-\frac{\pi}{6}=\frac{5\pi}{6}\approx\frac{5\times3.1416}{6}=\frac{15.708}{6}\approx2.62\) radians.

Answer:

2.62