To find all points where the tangent line to the graph of $f(x) = 2\sin(x) + \sin^2(x)$ is horizontal, we need to find the values of $x$ where the derivative of the function, $f'(x)$, equals zero.

Step 1: Differentiate the function

The given function is: $f(x) = 2\sin(x) + \sin^2(x)$

Now, differentiate $f(x)$ with respect to $x$:

1. The derivative of $2\sin(x)$ is: $\frac{d}{dx}[2\sin(x)] = 2\cos(x)$

2. The derivative of $\sin^2(x)$ (using the chain rule) is: $\frac{d}{dx}[\sin^2(x)] = 2\sin(x)\cos(x)$

Thus, the derivative of $f(x)$ is: $f'(x) = 2\cos(x) + 2\sin(x)\cos(x)$

Step 2: Set the derivative equal to zero

To find where the tangent line is horizontal, set $f'(x) = 0$:

$2\cos(x) + 2\sin(x)\cos(x) = 0$

Factor out $2\cos(x)$:

$2\cos(x)[1 + \sin(x)] = 0$

Step 3: Solve for $x$

For this product to equal zero, either $\cos(x) = 0$ or $1 + \sin(x) = 0$.

Case 1: $\cos(x) = 0$

The solutions to $\cos(x) = 0$ are: $x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}$ where $n$ is any integer.

Case 2: $1 + \sin(x) = 0$

Solve for $\sin(x)$: $\sin(x) = -1$ The solutions to $\sin(x) = -1$ are: $x = \frac{3\pi}{2} + 2n\pi, \quad n \in \mathbb{Z}$

Step 4: Combine the solutions

Thus, the points at which the tangent line is horizontal are given by: $x = \frac{\pi}{2} + n\pi \quad \text{or} \quad x = \frac{3\pi}{2} + 2n\pi$ where $n \in \mathbb{Z}$.

These are all the $x$-coordinates where the tangent line is horizontal. To find the corresponding points on the graph, substitute these values of $x$ into the original function $f(x)$.

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Would you like details on finding the actual points on the graph or explanations of any steps?

Here are 5 related questions:

1. How do you find the derivative of a trigonometric function?
2. What is the significance of the horizontal tangent line on a graph?
3. How would you solve other equations involving trigonometric identities?
4. Can you find the horizontal tangent points for the function $g(x) = 3\sin(x) + \cos^2(x)$?
5. What is the graphical interpretation of solving $\cos(x) = 0$?

Tip: Always check for factoring opportunities when solving equations involving products, as this can simplify solving significantly.