$$y=3\cos2x + 7\sin x +2$$
The original question was: Finding the stationary point of the function.
I tried to use both differentiation and trigonometric substitution of $\cos(2x)$ into $1-2\sin^2(x)$.
For the quadratic method, I substituted $\sin(x)$ as $u$ and solved for $u$, where the the equation is equated to $0$, and finding the midpoint of the roots to find out the value of $x$ following the nature of a parabolic equation.
Unfortunately I realized that trigonometric substitution eliminates a possible solution for $x$ where $\cos(x) =0$. I suspect that it might be due to a non function being converted into a function but I do not quite understand why does it causes so.
Best Answer
First impulse is to just differentiate.
$y=3\cos2x + 7\sin x +2 $ so
$\begin{array}\\ y' &=-6\sin(2x) + 7\cos x\\ &=-12\sin(x)\cos(x) + 7\cos x\\ &=\cos(x)(-12\sin(x) + 7)\\ \end{array} $
so $\cos(x) = 0$ or $\sin(x) = \dfrac{7}{12} $.