Tricky Logs

Geometry Level 1

$\left [ \log_{10} \left ( \tan \left ( 1^\circ \right ) \right ) \right ] \times \left [ \log_{10} \left ( \tan \left ( 2^\circ \right ) \right ) \right ] \times \left [ \log_{10} \left ( \tan \left ( 3^\circ \right ) \right ) \right ] \times \cdot \cdot \cdot \\ \times \left [ \log_{10} \left ( \tan \left ( 88^\circ \right ) \right ) \right ] \times \left [ \log_{10} \left ( \tan \left ( 89^\circ \right ) \right ) \right ] = \ ?$

The answer is 0.

2 solutions

Tijmen Veltman
Apr 8, 2015

For every $n=1,2,\ldots,89$ , $\tan(n^\circ)>0$ hence $\log(\tan(n^\circ))$ is defined. Since $\log(\tan 45^\circ)=\log(1)=0$ , the total product is also equal to $\boxed{0}$ .

Swarnendu Bhattacharjee
Jan 6, 2016

also log(tanx)*log(tan(90-x))=0....for any x so,the ans must be zero...

Wait.. Tan(1) is the reciprocal of tan (89) so log(x) log(1/x) = log (x) log(-x) is not 0. I think only the first answer stands. C.

Chris Crawford - 5 years, 5 months ago

No, log(tanx)*log(tan(90-x)) = log(tanx + 1/tanx)...

Canwen Jiao - 3 years ago

Tricky Logs

The answer is 0.

2 solutions

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