Answer
If the terminal side of an angle $\theta$ lies in quadrant III, then the value of tan $\theta$ and cot $\theta$ are positive, and all other trigonometric function values are negative.
Work Step by Step
Since both tan $\theta$ and cot $\theta$ are dependent of the values of (x, y) and in quadrant III both x and y are negative, using the law of signs we know that $\frac{-}{-} = +$.
In the case of all other trigonometric functions, the distance from the point of origin is necessary to find the value of the functions. Since the formula for this distance (r) is $r=\sqrt (x^{2} + y^{2})$, this value is always positive, so for all other trigonometric functions the case will always be $\frac{-}{+} = \frac{+}{-} = -$.
Therefore if the terminal side of an angle $\theta$ lies in quadrant III, then the value of tan $\theta$ and cot $\theta$ are positive, and all other trigonometric function values are negative.
