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
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.
Hint: $|\sin \alpha| \leq 1$.

Hence, this implies that $ c = \frac {43+(-37)} {2} $.