<s>[INST] Difficult trigonometry problem to find the minimum value Find the minimum value of $5\cos A + 12\sin A + 12$.
I don't  know how to approach this problem. I need help.
I'll show you how much I got...
$$5\cos A +12\sin A + 12 = 13(5/13\cos A +12/13\sin A) + 12$$
 [/INST] As we know that,
$−\sqrt{a^2+b^2} \le a \cosθ + b \sinθ \le \sqrt{a^2+b^2}$
So for $5 \cos θ+12 \sin θ$ we have,
$−\sqrt{5^2+12^2} \le 5cosθ+12sinθ \le \sqrt{5^2 +12^2}$ 
= $−13 ≤ 5cosθ +12sinθ ≤ 13$
Now adding 12 we get, 
= $−13 + 12 ≤ 5 \cosθ + 12 \sinθ + 12 ≤ 13 + 12$
= $− 1 ≤ 5 \cosθ + 12 \sinθ + 12 ≤ 25$
So minimum value = −1
And maximum value = 25
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