Finding the minimum value of $\frac{\mu\cos\theta}{\cos\theta + \mu\sin\theta}$

Question

This is a part of a physics problem I was solving. I was supposed to find the minimum value of $\dfrac{\mu\cos\theta}{\cos \theta +\mu\sin \theta}$ where $\mu= 0.2$

The solution given used calculus to determine the minimum value (by finding the first derivative). However, being a pre-calculus student, I wonder if there's a way to solve this problem without using derivatives. I tried using $AM/GM$ but there's a $\mu$ causing trouble.

Any method to solve this without calculus?

Answer 1

Originally Answered: Find min-max of $\cos{\theta + \mu\sin{\theta}}$

The general form of the answer is

$$ -\sqrt{a^2+b^2} \leq a\cos{\theta} + b\sin{\theta} \leq \sqrt{a^2+b^2} $$

Look for the non calculus solution below.

Does this help?

This article gives a nice, non calculus proof.

The Edited Question (totally different): Find min-max of $\frac{\mu\cos{\theta}}{\cos{\theta + \mu\sin{\theta}}}$

Dividing numerator and denominator by $\cos{\theta}$ we get,

$$ \frac{\mu}{1+\mu\tan{\theta}} $$

Note that the numerator is constant while the denominator has only one variable $\tan{\theta}$ term.

Finding a min-max value for the function is futile, because as this graph shows, the function can achieve any real value and is not bounded.