Determine the coordinates of a point on the flipped unit circle given certain conditions

Let’s consider the unit circle equation flipped along y=x: $x^2 + y^2 = 1$ becomes $y = x \cdot \sqrt{1 – x^2}$. Given a point where the x-coordinate is $\frac{1}{2}$, find the corresponding y-coordinate.

Since the point lies on the flipped unit circle, we have:

$y = \frac{1}{2} \cdot \sqrt{1 – (\frac{1}{2})^2}$

$y = \frac{1}{2} \cdot \sqrt{1 – \frac{1}{4}}$

$y = \frac{1}{2} \cdot \sqrt{\frac{3}{4}}$

$y = \frac{1}{2} \cdot \frac{\sqrt{3}}{2}$

$y = \frac{\sqrt{3}}{4}$

Hence, the point on the flipped unit circle is $\left(\frac{1}{2}, \frac{\sqrt{3}}{4}\right)$.

Let us first acknowledge that the given flipped unit circle equation is $y = x cdot sqrt{1 – x^2}$. For the x-coordinate $frac{1}{2}$, we need to calculate the y-value.

Using the equation:

$y = frac{1}{2} cdot sqrt{1 – (frac{1}{2})^2}$

$y = frac{1}{2} cdot sqrt{frac{3}{4}}$

$y = frac{1}{2} cdot frac{sqrt{3}}{2}$

$y = frac{sqrt{3}}{4}$

Therefore, the coordinates are $left(frac{1}{2}, frac{sqrt{3}}{4}
ight)$.

Given $y = x cdot sqrt{1 – x^2}$ and $x = frac{1}{2}$, find $y$:

$y = frac{1}{2} cdot sqrt{frac{3}{4}}$

$y = frac{sqrt{3}}{4}$

Thus, the coordinates are $left(frac{1}{2}, frac{sqrt{3}}{4}
ight)$.

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