5 sin(x)+ 12 cos(x) + r is always greater than or equal to zero. What is the smallest value satisfied by 'r' ?

Given,

 5sinx + 12cosx ≥ -r 

$\Rightarrow 13\frac{5}{13}\sin x + \frac{12}{13}\cos x \ge -r$cosx) ≥ -r 

$\frac{5}{13}$= cosA => sinA = $\frac{12}{13}$

=> 13(sinx cosA + sinA cosx) ≥ -r

=> 13(sin(x + A)) ≥ -r

we know that , -1 ≤ sin (angle) ≤ 1

=> 13sin (x + A) ≥ -13 $\Rightarrow rmin =13$