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To solve for the equation of the tangent line to the curve \( y = \sin(\sin(x)) \) at the given point \( (2\pi, 0) \), we follow these steps:

### 1. Find the derivative of \( y = \sin(\sin(x)) \)
We need to compute the derivative \( \frac{dy}{dx} \). Using the chain rule, we get:
\[
\frac{dy}{dx} = \cos(\sin(x)) \cdot \cos(x)
\]
This derivative will give us the slope of the tangent line at any point \( x \).

### 2. Evaluate the derivative at \( x = 2\pi \)
Now, substitute \( x = 2\pi \) into the derivative to find the slope of the tangent line at that point:
\[
\frac{dy}{dx} \Big|_{x=2\pi} = \cos(\sin(2\pi)) \cdot \cos(2\pi)
\]
Since \( \sin(2\pi) = 0 \) and \( \cos(2\pi) = 1 \), we have:
\[
\frac{dy}{dx} \Big|_{x=2\pi} = \cos(0) \cdot 1 = 1
\]
So, the slope of the tangent line at \( (2\pi, 0) \) is 1.

### 3. Use the point-slope form to find the equation of the tangent line
The equation of a line is given by the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency. Substituting \( m = 1 \), \( x_1 = 2\pi \), and \( y_1 = 0 \), we get:
\[
y - 0 = 1(x - 2\pi)
\]
which simplifies to:
\[
y = x - 2\pi
\]

### Final Answer:
The equation of the tangent line is \( y = x - 2\pi \).

Would you like more details or have any other questions?

Here are 5 related questions to expand your understanding:
1. How would the derivative change if the function were \( y = \cos(\sin(x)) \)?
2. Can the tangent line ever be horizontal for the function \( y = \sin(\sin(x)) \)?
3. What is the second derivative of \( y = \sin(\sin(x)) \), and what does it represent geometrically?
4. How does the chain rule apply to more complicated compositions of trigonometric functions?
5. How do you find the tangent line to a parametric curve?

**Tip:** Always remember to check the values of trigonometric functions at key points like \( 0, \pi, 2\pi \), as they can often simplify your calculations!

Find an equation of the tangent line to the curve y = sin(sin(x)) at the point (2π, 0).

This problem provides a step-by-step solution for finding the equation of the tangent line to the curve y = sin(sin(x)) at the point (2π, 0) using derivatives and the chain rule. Suitable for high school calculus students.

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