How to find an appropriate constant value to which the function will be defined

Question

I have this function $y=\frac{-\ln(2\cos{t}+e^{2k}-2)}{2}$, and i'm trying to find a value for k for which $y$ is defined for all $t$.

So i need the thing in the $ln$ to be larger than zero, i see that this inequality should hold $2\cos(t) +e^{2k}>2$.

How do i continue form here to reach that if $k$ is larger than some constant then $y$ is defined for every $t$?

Answer 1

You should recall that minimum value of cos is -1. So for "worst" scenario you have $$2*(-1)+e^{2k}>2$$ or $$e^{2k}>4$$ Then, ln both sides: $2k>\ln4$, so $k>0.5 \ln4$

Answer 2

Hint $\cos t$ is bounded, so all you need is to find a $k$ s.t. $e^{2k}>4$… Taking logs seem the next obvious step...

Answer 3

Hint: Recall that $$-1 \leq \cos (t) \leq 1$$

What is the least possible value for $2\cos (t)$ then?