Questions: Given a unit circle, what is the value of x at the indicated point? x=[?]√[ ] Hint: The equation for the unit circle is x^2+y^2=1.

Solution Steps

The problem provides a unit circle with a point \((x, -\frac{5}{7})\) on it. The equation of the unit circle is \(x^2 + y^2 = 1\).

Substitute \(y = -\frac{5}{7}\) into the equation \(x^2 + y^2 = 1\): \[ x^2 + \left(-\frac{5}{7}\right)^2 = 1 \]

Calculate \(\left(-\frac{5}{7}\right)^2\): \[ \left(-\frac{5}{7}\right)^2 = \frac{25}{49} \] So the equation becomes: \[ x^2 + \frac{25}{49} = 1 \]

Subtract \(\frac{25}{49}\) from both sides of the equation: \[ x^2 = 1 - \frac{25}{49} \] Convert 1 to a fraction with a denominator of 49: \[ x^2 = \frac{49}{49} - \frac{25}{49} \] \[ x^2 = \frac{24}{49} \]

Take the square root of both sides: \[ x = \pm \sqrt{\frac{24}{49}} \] \[ x = \pm \frac{\sqrt{24}}{7} \] Simplify \(\sqrt{24}\): \[ \sqrt{24} = 2\sqrt{6} \] So: \[ x = \pm \frac{2\sqrt{6}}{7} \]

Final Answer