<s>[INST] Find the values of the positive constants $k$ and $c$ such that $-37\le k(3\sin\theta + 4\cos\theta) +c\le 43$ for all values of $\theta$ Hi how do i go about solving this?
Find the values of the positive constants $k$ and $c$ such that $$-37\le k(3\sin\theta + 4\cos\theta) +c\le 43 $$for all values of $\theta$ $$\rightarrow-37\le k(5(\sin\theta + 53.1)) +c\le 43 $$ 
Then what?
Cheers
 [/INST] By Cauchy-Schwarz
$$(3\sin(x)+4\cos(x))^2 \leq 25 (\sin^2(x)+\cos^2(x))=25$$
and equality is possible.
Then 
$$-5 \leq 3\sin(x)+4\cos(x) \leq 5 $$
this shows that
$$-5k+c \leq - k(3\sin\theta + 4\cos\theta) +c\le 5k+c \,.$$
and the lower/upper bounds can be atatined. You can finish it easely.
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