Minimum value of trigonometry function How can I get the minimum value of function
$$f(x) = (2 + \sin x)(5 - \sin x)$$
I have used the differential ways but the answer was not match with the key answer.
By the way the key answer is $6$.

You can use the change of variable $t=\sin x$ and remember that $-1\le t\le 1$.
Then
$$(2+t)(5-t)=10+3t-t^2$$
has the derivative $3-2t$, which cancels at $t=\frac32$, out of the domain of $t$.
The minimum can only arise at one of the ends of that domain,
$$f(-1)=\color{green}6$$or $$f(1)=12.$$
No trigonometry !
