In which quadrant are all functions negative in the unit circle? Is it even possible?

In the unit circle, functions can be evaluated based on the coordinates of points in different quadrants. The unit circle is divided into four quadrants:

  • Quadrant I: Both x and y coordinates are positive (0° to 90°).
  • Quadrant II: x is negative and y is positive (90° to 180°).
  • Quadrant III: Both x and y coordinates are negative (180° to 270°).
  • Quadrant IV: x is positive and y is negative (270° to 360°).

When we refer to functions, such as sine, cosine, and tangent, their signs depend on the quadrant:

  • In Quadrant I, all functions are positive.
  • In Quadrant II, sine is positive, while cosine and tangent are negative.
  • In Quadrant III, sine and cosine are negative, and tangent is positive.
  • In Quadrant IV, sine is negative, while cosine and tangent are positive.

From this analysis, we can conclude that there is no quadrant in the unit circle where all trigonometric functions are negative. In Quadrant III, while sine and cosine are negative, tangent becomes positive since it is the ratio of sine to cosine. Therefore, it is not possible for all functions to be negative in any quadrant of the unit circle.

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