University of South Carolina High School Math Contest/1993 Exam/Problem 18
The minimum value of the function
as varies over all numbers in the largest possible domain of , is
Recall the trigonometric identities
Now we must consider the quadrant that is in. If is in quadrant I, then all of the trig functions are positive and . If is in quadrant II, then sine is positive and the rest of cosine, tangent, and cotangent are negative giving . If is in quadrant III, then tangent and cotangent are positive while sine and cosine are negative, making . Finally, if is in quadrant IV, then only cosine is positive with the other three being negative giving . Thus our answer is -2.