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Geometry Level 2

If $x$ is an integer in between 0 and 180 inclusive such that $\tan(x^\circ) \ne 0$ , find the value of $x$ such that $\cot(x^\circ)$ is maximized.

1 solution

Tom Engelsman
May 22, 2021

Knowing that $f(x) = \tan(\frac{\pi x}{180})$ :

1) Is an increasing function over $[0,90) \cup (90,180]$ ;

2) Is zero at $x = 0, 180;$

3) Is positive valued over $(0,90)$ ;

4) Is negative valued over $(90,180)$

The function $g(x) = \cot(\frac{\pi x }{180})$ is maximum at $\boxed{x=1}.$