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1987 AMC 12 #20 - Logs and Trig

Source:

December 31, 2011
trigonometrylogarithmsAMC

Problem Statement

Evaluate log⁡10(tan⁡1∘)+log⁡10(tan⁡2∘)+log⁡10(tan⁡3∘)+⋯+log⁡10(tan⁡88∘)+log⁡10(tan⁡89∘). \log_{10}(\tan 1^{\circ})+ \log_{10}(\tan 2^{\circ})+ \log_{10}(\tan 3^{\circ})+ \cdots + \log_{10}(\tan 88^{\circ})+\log_{10}(\tan 89^{\circ}). <spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 12log⁡10(32)<spanclass=′latex−bold′>(C)</span> 12log⁡102<spanclass=′latex−bold′>(D)</span> 1<spanclass=′latex−bold′>(E)</span> none of these <span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ \frac{1}{2}\log_{10}(\frac{\sqrt{3}}{2}) \qquad<span class='latex-bold'>(C)</span>\ \frac{1}{2}\log_{10}2 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ \text{none of these}
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