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On a unit circle, the vertical distance from the x-axis to a point on the perimeter of the circle is twice the horizontal distance
from the y-axis to the same point. What is sin theta?

On a unit circle the vertical distance from the xaxis to a point on the perimeter of the circle is twice the horizontal distance from the yaxis to the same poin class=

Respuesta :

Check the picture below.

since the vertical distance, namely the y-coordinate, is twice as much as the horizontal, then if the horizontal is "x", the vertical one must be 2x.

let's find the hypotenuse first.

[tex]\bf \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=\sqrt{a^2+b^2} \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{x}\\ b=\stackrel{opposite}{2x}\\ \end{cases} \\\\\\ c=\sqrt{x^2+(2x)^2}\implies c=\sqrt{x^2+4x^2}\implies c=\sqrt{5x^2}\implies c=x\sqrt{5} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf sin(\theta )=\cfrac{\stackrel{opposite}{2~~\begin{matrix} x \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ }}{\stackrel{hypotenuse}{~~\begin{matrix} x \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ \sqrt{5}}}\implies \stackrel{\textit{and rationalizing the denominator}~\hfill }{\cfrac{2}{\sqrt{5}}\cdot \cfrac{\sqrt{5}}{\sqrt{5}}\implies \cfrac{2\sqrt{5}}{(\sqrt{5})^2}\implies \cfrac{2\sqrt{5}}{5}}[/tex]

Ver imagen jdoe0001
ashishdwivedilVT

The value of Sin θ for a given circle and given condition is (2√5)/5.

What is Circle?

A circle is the set of all points in the plane that are a fixed distance (the radius) from a fixed point (the center). Any interval joining a point on the circle to the center is called a radius.

Here, let the horizontal side be x.

         vertical side be y

then as per the given condition

Vertical distance = 2 X horizontal distance

      y = 2x

By Pythagoras Theorem,

      r = [tex]\sqrt{x^2+y^2}[/tex]

      1 = [tex]\sqrt{x^2+(2x)^2}[/tex]

      1 = x² + 4x²

      5x² = 1

       x² = 1/5

      x = 1/√5

then, y = 2/√5

Now, Sin θ = perp. / Hypo.

         Sin θ = y/1

           Sin θ  = 2/√5

or        Sin θ  = (2√5)/5

Thus, the value of Sin θ for a given circle and given condition is (2√5)/5.

Learn more about Circle from:

https://brainly.com/question/11833983

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