Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 18"
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<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> | ||
− | Now we must consider the quadrant that <math>x</math> is in. If <math>x</math> is in quadrant I, then all of the trig functions are positive and <math>f(x)=1+1+1+1=4</math>. If <math>x</math> is in quadrant II, then sine is positive and the rest of cosine, tangent, and cotangent are negative giving <math>f(x)=1-1-1-1=-2</math>. If <math>x</math> is in quadrant III, then tangent and cotangent are positive while sine and cosine are negative making <math>f(x)=1+1-1-1=0</math>. Finally, if <math>x</math> is in quadrant IV, then only cosine is positive with the other three being negative giving <math>f(x)=-1+1-1-1=-2</math>. Thus our answer is -2. | + | Now we must consider the quadrant that <math>x</math> is in. If <math>x</math> is in quadrant I, then all of the trig functions are positive and <math>f(x)=1+1+1+1=4</math>. If <math>x</math> is in quadrant II, then sine is positive and the rest of cosine, tangent, and cotangent are negative giving <math>f(x)=1-1-1-1=-2</math>. If <math>x</math> is in quadrant III, then tangent and cotangent are positive while sine and cosine are negative, making <math>f(x)=1+1-1-1=0</math>. Finally, if <math>x</math> is in quadrant IV, then only cosine is positive with the other three being negative giving <math>f(x)=-1+1-1-1=-2</math>. Thus our answer is -2. |
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Revision as of 13:20, 12 October 2007
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.