Help finding all points on a graph of the function $f(x)= 2 \sin(x) + \sin^2(x)$ at which the tangent line is horizontal?

Question

I would appreciate help finding the answer to this question :]. Again the question is,

Find all the points on the graph of the function:

$f(x)= 2 \sin(x) + \sin^2(x)$

the second trig function is supposed to be sin squared of x but not sure how to make that on here...

at which the tangent line is horizontal.

Answer 1

We can calculate $$f'(x)=2\cos(x)+2\sin(x)\cos(x)=2\cos(x)(1+\sin(x)).$$ Now, the tangent is horizontal whenever $f'(x)=0$, so this means when $$2\cos(x)=0\quad\text{or}\quad1+\sin(x)=0.$$ Solving the cosine equation gives $x=\pi/2+k\pi$ for $k\in\Bbb Z$ while solving the sine equation gives $-\pi/2+2k\pi$. Since the solution for the sine equation is actually contained in the set of solutions for the cosine equation, we can write our final answer as $x=\pi/2+k\pi$ for $k\in\Bbb Z$.