Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 18"
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We can now simplify the function to | We can now simplify the function to | ||
− | <center><math> f(x) = \frac{\sin(x)}{ | + | <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. | |
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Revision as of 20:00, 23 July 2006
Problem
The minimum value of the function
![$\displaystyle f(x) = \frac{\sin (x)}{\sqrt{1 - \cos^2 (x)}} + \frac{\cos(x)}{\sqrt{1 - \sin^2 (x) }} + \frac{\tan(x)}{\sqrt{\sec^2 (x) - 1}} + \frac{\cot (x)}{\sqrt{\csc^2 (x) - 1}}$](http://latex.artofproblemsolving.com/c/6/a/c6a4e021d142f4bad5a913725ac6e8f1316b9066.png)
as varies over all numbers in the largest possible domain of
, is
![$\mathrm{(A) \ }-4 \qquad \mathrm{(B) \ }-2 \qquad \mathrm{(C) \ }0 \qquad \mathrm{(D) \ }2 \qquad \mathrm{(E) \ }4$](http://latex.artofproblemsolving.com/2/f/7/2f7612b8e989eded4129ff00a31811dee6a6ef5e.png)
Solution
Recall the Pythagorean Identities:
![$\sin^2 x + \cos^2 x = 1$](http://latex.artofproblemsolving.com/a/9/5/a958fb1f7b5b87ef62ed4a2205643ef36cb3df4d.png)
![$\tan^2 x + 1 = \sec^2 x$](http://latex.artofproblemsolving.com/5/3/f/53fffbe6880b6565c56fbfa616bcbf130036e766.png)
![$1 + \cot^2 x = \csc^2 x$](http://latex.artofproblemsolving.com/2/2/b/22b3b506380cec4206c8a8c63c9397a1295d990d.png)
We can now simplify the function to
![$f(x) = \frac{\sin(x)}{|\sin (x)|}+\frac{\cos(x)}{\|\cos(x)|} + \frac{\tan(x)}{|\tan(x)|} + \frac{\cot(x)}{|\cot(x)|}.$](http://latex.artofproblemsolving.com/f/d/6/fd689aa74c0645477b01a8ef8a4e307e610c6f8a.png)
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.