on a unit circle, the vertical distance from the x - axis to a point on the perimeter of the circle is twice…

Explanation:

Step1: Define the coordinates

Let the point on the unit - circle be $(x,y)$. The vertical distance from the $x$ - axis to the point is $|y|$, and the horizontal distance from the $y$ - axis to the point is $|x|$. Given $|y| = 2|x|$.

Step2: Use the unit - circle equation

The equation of a unit - circle is $x^{2}+y^{2}=1$. Substitute $y = 2x$ or $y=-2x$ into the unit - circle equation. If $y = 2x$, then $x^{2}+(2x)^{2}=1$, which simplifies to $x^{2}+4x^{2}=1$, so $5x^{2}=1$, and $x^{2}=\frac{1}{5}$, $x=\pm\frac{1}{\sqrt{5}}$. Then $y = 2x=\pm\frac{2}{\sqrt{5}}$. If $y=-2x$, then $x^{2}+(-2x)^{2}=1$, which also gives $5x^{2}=1$, $x = \pm\frac{1}{\sqrt{5}}$ and $y=\mp\frac{2}{\sqrt{5}}$.

Step3: Recall the definition of sine

For a point $(x,y)$ on the unit - circle, $\sin\theta=y$. Since $\sin\theta$ represents the $y$ - coordinate of a point on the unit - circle, and $y=\pm\frac{2}{\sqrt{5}}=\pm\frac{2\sqrt{5}}{5}$.

Answer:

$\pm\frac{2\sqrt{5}}{5}$