Tangen

Geometry Level 2

paragraph 1 What is the value of log tan 1^{o} + log tan 2^{o} + log tan 3^{o} + .... + log tan 89^{o} .

0 -1 1 2

2 solutions

Christopher Serino
Dec 31, 2016

First, $\log(x) + \log(y) = \log xy$ .

Then, $\tan(90^\circ - x) = \cot x$

Of course $\tan(x)\cot(x)=1$

Therefore $\tan(1^\circ) \tan(89^\circ)=1$ and similarly for 2 and 88, 3 and 87, and any and all other x and y that sum to a right angle.

Finally $\log 1 = 0$

I enjoyed this problem. It's like a trigonometric version of the famous Gauss anecdote .

I wanted to point out, though, that there's a more widely accepted and less ambiguous way to write your degree signs: a superscript \circ , as in $45^\circ$ . When your LaTeX compiles properly, the difference between $45^{o}$ and $45^\circ$ is very slight. But if it doesn't compile correctly, nobody's left wondering what "1 to the power of o" means. :)

Michael Hartwell - 4 years, 4 months ago

Nivedita Sharma
Feb 18, 2017

log x + log y =log xy So, log(tan1 tan 2 ........tan89), Now, as = tan(90 -1)=cot 1, And tan1.cot1=1 All will get cancelled except tan45 So, log (tan45) = log1 = 0

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