Finding the minimum value of $(\alpha+5-3|\cos \beta|)^2+(\alpha-2|\sin \beta|)^2$ We assume $\alpha,\beta$ are real numbers, I would like to find the minimum value of 

$$(\alpha+5-3|\cos \beta|)^2+(\alpha-2|\sin \beta|)^2.$$

After expanding $(\alpha+5)^2+9\cos^2 \beta-2(\alpha+5)|3\cos \beta|+\alpha^2+4\sin^2 \beta-4\alpha|2\sin \beta|$, I would have some help with this.


Find the minimum value of $(\alpha+5-3|\cos \beta|)^2+(\alpha-2|\sin \beta|)^2$

Hint. By expanding as you did then factoring the quadratic expression in $\alpha$ one gets
$$ \small
\begin{align}
(\alpha+5-3|\cos \beta|)^2+(\alpha-2|\sin \beta|)^2&=2\left(\alpha+\frac{5-3|\cos \beta|-2|\sin \beta|}2\right)^2 +\frac{\left(5-3|\cos \beta|+2|\sin \beta|\right)^2}2
\\\\&\ge\frac{\left(5-3|\cos \beta|+2|\sin \beta|\right)^2}2
\\\\&\ge 2.
\end{align}
$$
