<s>[INST] Max and min of $5 \sin\left(\frac{\pi}{6} x\right)+10 \cos\left(\frac{\pi}{6}x\right) +11.2$ without graphing 
$$P(x)= 5 \sin\left(\frac{\pi}{6} x\right)+10 \cos\left(\frac{\pi}{6} x\right) +11.2$$
  How would you mathematically find the the max and min points of $P(x)$ without graphing?

I know that individually if say $5 \sin\left(\frac{\pi}{6} x\right)+11.2$ is of $c(x)$ then max and min would be when $\sin\left(\frac{\pi}{6} x\right)= 1$ or $-1$ ( max = 16.2 min = 6.2)
thanks in advance
 [/INST] Assume that $$ \cos\ t=\frac{5}{\sqrt{5^2+10^2}},\ \sin\ t = \frac{10}{\sqrt{5^2+10^2}} $$
Hene $$ P(t)=\sqrt{5^2+10^2} \{ \cos\ t\sin\ \frac{\pi x}{6} + \sin\ t\cos\ \frac{\pi x}{6} \} + 11.2 $$ $$=5\sqrt{5}  \sin\ (\frac{\pi x}{6} + t)  + 11.2 $$
That is max$ =5\sqrt{5} +11.2$ and min$=-5\sqrt{5}+11.2 $
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