Prove that arctan(4*pi/3), in radians, is irrational.

arctan(a)=b implies that a is a ratio of the form y/x and b is an angle.  Specifically it the the angle formed by the positive x-axis and the ray from (0,0) through (x,y).

It is rather strange that the given ratio is an expression containing pi, usually it is the angle that would be a rational multiple of pi.

As such I would surprised to see that the arctangent of any rational multiple of pi was rational.

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The problem does not ask for tan(4pi/3) which is clearly an irrational number:  the negative square root of 2.

Posted by Jer on 2011-05-02 15:11:04