High School

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A point \( P \) is on the unit circle corresponding to an angle of \( t \). If \( \cos(t) = \frac{13}{15} \) and \( P \left( \frac{13}{15}, y \right) \) is in quadrant IV, then \( y = \)

Answer :

The y-coordinate of point P on the unit circle in quadrant IV with cos(t) = (13/15) is -2 * sqrt(14)/15.

To find the y-coordinate of point P that lies on the unit circle in quadrant IV, where cos(t) = (13/15), we must use the Pythagorean identity sin^2(t) + cos^2(t) = 1 for a unit circle. Since P lies in quadrant IV, the sin(t) must be negative because sine is negative in that quadrant.

First, we calculate sin^2(t):

  • sin^2(t) = 1 - cos^2(t) = 1 - (13/15)^2 = 1 - (169/225) = (225/225) - (169/225) = 56/225

Then we find sin(t):

  • sin(t) = -sqrt(56/225)
  • sin(t) = -sqrt(56)/15
  • sin(t) = -sqrt(4 * 14)/15
  • sin(t) = -2 * sqrt(14)/15

Thus, the y-coordinate of point P is -2 * sqrt(14)/15.

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