in the xy - plane, the terminal ray of an angle of measure \\( \\theta \\) in standard position intersects…

Explanation:

Step1: Recall the unit - circle definitions

For a point (P(x,y)) on the unit circle (x^{2}+y^{2}=1). Given that (x = 2y).

Step2: Substitute (x = 2y) into the unit - circle equation

Substitute (x) in (x^{2}+y^{2}=1), we get ((2y)^{2}+y^{2}=1). Expand the left - hand side: (4y^{2}+y^{2}=1), so (5y^{2}=1), then (y^{2}=\frac{1}{5}), and (y=\frac{1}{\sqrt{5}}) (since (0\leq\theta\leq\frac{\pi}{2}), (y\geq0)). And (x = 2y=\frac{2}{\sqrt{5}}).

Step3: Use the tangent function

We know that (\tan\theta=\frac{y}{x}). Substitute (x=\frac{2}{\sqrt{5}}) and (y = \frac{1}{\sqrt{5}}) into (\tan\theta), we have (\tan\theta=\frac{\frac{1}{\sqrt{5}}}{\frac{2}{\sqrt{5}}}=\frac{1}{2}). Then (\theta=\arctan(\frac{1}{2})). Using a calculator, (\arctan(\frac{1}{2})\approx0.464).

Answer:

A. 0.464