Minimum + Trigonometry = Minimometry!

Geometry Level 3

Consider this function; $f(x)=\min{(\tan x, \cot x)}$

Then for what values of $x$ does the output of the function be 1?

Clarification: In the options, $n$ is an integer.

None of these choices

n\pi + \dfrac{\pi}{6}

\dfrac{n\pi }{2}+ \dfrac{\pi}{4}

n\pi + \dfrac{\pi}{3}

n\pi + \dfrac{\pi}{2}

n\pi + \dfrac{\pi}{4}

n\pi + \dfrac{\pi}{8}

1 solution

Tom Engelsman
Jun 9, 2018

The functions $tan(x), cot(x)$ are positive in the FIRST & the THIRD quadrants in the xy-plane, and they both equal 1 for:

$x = \frac{\pi}{4}; \frac{5\pi}{4}; \frac{9\pi}{4}; \frac{13\pi}{4}; .... ; \frac{(4n+1)\pi}{4}$ for $n \in \mathbb{Z}$

or when $\boxed{x = n\pi + \frac{\pi}{4}}.$