Root of Trig

Geometry Level pending

If the maximum value of the function $f(x)=a\sqrt{(\sin x+2)^2}+4\ (a>0)$ is $10,$ what is the value of $f\left(\frac{\pi}{6}\right)?$

9 8 7 6

1 solution

Tom Engelsman
Nov 10, 2020

The maximum value of $f(x)$ occurs when the quantity $\sin(x) = 1$ , or:

$10 = a\sqrt{(1+2)^2} + 4 = a|3| + 4 \Rightarrow a = 2$ .

Thus, $f(\frac{\pi}{2}) = 2|\sin(\frac{\pi}{6})+2| + 4 = 2|\frac{1}{2}+2|+4 = |5|+4 = \boxed{9}.$