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The point P=(x,(2)/(3)) lies on the unit circle shown below. What is the value of x in simplest form? (Note: the figure is not drawn to scale)

The point P=(x,23) P=\left(x, \frac{2}{3}\right) lies on the unit circle shown below. What is the value of x x in simplest form?\newline(Note: the figure is not drawn to scale)

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Full solution

Q. The point P=(x,23) P=\left(x, \frac{2}{3}\right) lies on the unit circle shown below. What is the value of x x in simplest form?\newline(Note: the figure is not drawn to scale)
  1. Apply Pythagorean Identity: Since PP lies on the unit circle, we use the Pythagorean identity: x2+y2=1x^2 + y^2 = 1.
  2. Substitute yy with 23\frac{2}{3}: Substitute yy with 23\frac{2}{3}: x2+(23)2=1x^2 + \left(\frac{2}{3}\right)^2 = 1.
  3. Calculate (23)2(\frac{2}{3})^2: Calculate (23)2(\frac{2}{3})^2: x2+49=1x^2 + \frac{4}{9} = 1.
  4. Subtract 49\frac{4}{9}: Subtract 49\frac{4}{9} from both sides: x2=1−49x^2 = 1 - \frac{4}{9}.
  5. Simplify right side: Simplify the right side: x2=99−49.x^2 = \frac{9}{9} - \frac{4}{9}.
  6. Continue simplifying: Continue simplifying: x2=59x^2 = \frac{5}{9}.
  7. Take square root: Take the square root of both sides: x=±59x = \pm\sqrt{\frac{5}{9}}.
  8. Simplify square root: Simplify the square root: x=±5/3x = \pm\sqrt{5}/3.

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