Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 18"
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− | + | Since <math>\sqrt{x^2} = |x|</math> for [[real number|real]] <math>x</math>, we can now simplify the [[function]] to | |
<center><math> f(x) = \frac{\sin(x)}{|\sin (x)|}+\frac{\cos(x)}{|\cos(x)|} + \frac{\tan(x)}{|\tan(x)|} + \frac{\cot(x)}{|\cot(x)|}. </math></center> | <center><math> f(x) = \frac{\sin(x)}{|\sin (x)|}+\frac{\cos(x)}{|\cos(x)|} + \frac{\tan(x)}{|\tan(x)|} + \frac{\cot(x)}{|\cot(x)|}. </math></center> |
Revision as of 15:40, 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
Since for real , 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.