# 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?

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.

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You might find this useful if you're curious how to use math font on MSE: MathJax Primer – gnometorule Feb 20 '13 at 4:31
Awesome! Thank you! :] – bkaifos15 Feb 20 '13 at 4:32
The tangent is horizontal if and only if $f'(x)=$? – 1015 Feb 20 '13 at 4:33
What I posted is the entire question...it doesn't ask/say anything else. – bkaifos15 Feb 20 '13 at 4:36
That's why we are asking you, what do you know about when the graph of a function is horizontal? – Gerry Myerson Feb 20 '13 at 5:01

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 $k\in\Bbb Z$.