How are the signs of Sin, Cos, and Tan represented in Quadrant III?

In Quadrant III of the Cartesian coordinate system, both the x-coordinates and y-coordinates of points are negative. When evaluating the sine, cosine, and tangent functions in this quadrant, we can determine their signs based on the definitions of these trigonometric functions.

1. The sine function, which measures the ratio of the opposite side to the hypotenuse in a right triangle, is associated with the y-coordinate. Since the y-coordinate is negative in Quadrant III, the sine of any angle in this quadrant will also be negative.

2. The cosine function measures the ratio of the adjacent side to the hypotenuse. In Quadrant III, the x-coordinate is negative, which means the cosine of any angle in this quadrant will be negative as well.

3. The tangent function is the ratio of the sine to the cosine (tan = sin/cos). With both sine and cosine being negative in Quadrant III, their ratio (tangent) will be positive, as a negative divided by a negative results in a positive.

Thus, for Quadrant III, the correct representation is that sine is negative, cosine is negative, and tangent is positive. This aligns perfectly with the information given in the correct choice.