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University of South Carolina High School Math Contest/1993 Exam/Problem 10

Problem

$\arcsin\left(\frac{1}{3}\right) + \arccos\left(\frac{1}{3}\right) + \arctan\left(\frac13\right) + \text{arccot}\left(\frac13\right) =$

$\mathrm{(A) \ }\pi \qquad \mathrm{(B) \ }\pi/2 \qquad \mathrm{(C) \ }\pi/3 \qquad \mathrm{(D) \ }2\pi/3 \qquad \mathrm{(E) \ }3/\pi/4$

Solution

If we construct right triangles for each pair of arguments ($\arcsin, \arccos$ in one triangle and $\arctan, \text{arccot}$ in another), we see that the sum of the angles is $90^\circ+90^\circ=\pi$.

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