Question

Find the points on the curve below at which the tangent is horizontal. Use n as an arbitrary integer. (Enter your answers as a comma-separated list.) y=cosx/(2+sinx)

Answer 1

The tangent line is horizontal when its slope is 0, which is the same as when the derivative is 0.
\[f(x)=\frac{\cos x}{2+\sin x}~~~\Rightarrow~~~f'(x)=\frac{-\sin x(2+\sin x)-\cos^2x}{(2+\sin x)^2}\]
The derivative will be zero when its numerator is zero:
\[\begin{align*}-\sin x(2+\sin x)-\cos^2x&=0\\
-2\sin x-\sin^2x-(1-\sin^2x)&=0\\
-2\sin x-1&=0\\
\sin x&=-\frac{1}{2}\\
x&=\frac{7\pi}{6}+2n\pi,~\frac{11\pi}{6}+2n\pi
\end{align*}\]