Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 18"
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<center><math> \sin^2 x + \cos^2 x = 1 </math> </center> | <center><math> \sin^2 x + \cos^2 x = 1 </math> </center> |
Revision as of 12:13, 31 July 2006
Problem
The minimum value of the function
as varies over all numbers in the largest possible domain of , is
Solution
Recall the trigonometric identities
We can now simplify the function to
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.