Evaluate &log_10(tan1°) & +log_10(tan2°) & +log_10(tan3°)+… & +log_10(tan88°) & +log_10(tan89°).

1987 AMC 12 Problem 20

Attempt Problem 20 of the 1987 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1987 AMC 12 solutions, or check the answer key.

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20.

Evaluate

\begin{aligned} &\log_{10}(\tan1^\circ)\\ &\quad+\log_{10}(\tan2^\circ)\\ &\quad+\log_{10}(\tan3^\circ)+\cdots\\ &\quad+\log_{10}(\tan88^\circ)\\ &\quad+\log_{10}(\tan89^\circ). \end{aligned}

0

\frac12\log_{10}\left(\frac{\sqrt3}{2}\right)

\frac12\log_{10}2

1

none of these

Pair the term at angle

k^\circ

with the term at angle

(90-k)^\circ

Use

\tan(90^\circ-\theta)=\cot\theta

before combining logarithms

For each

k=1,\ldots,44,

\tan k^\circ\tan(90^\circ-k^\circ)=1.

Thus each paired sum of logarithms is

\log_{10}1=0.

The remaining middle term is

\log_{10}(\tan45^\circ)=0.

Hence the whole sum is

0.

Thus the correct answer is A.