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We are asked to find the equation of the tangent line to the function \( y = f(x) \) at \( x = \frac{3\pi}{4} \), where \( f(x) = -\sin(x) \).

### Step 1: Find the derivative of \( f(x) \)
The derivative of \( f(x) = -\sin(x) \) will give us the slope of the tangent line at any point \( x \). 

\[
f'(x) = -\cos(x)
\]

### Step 2: Calculate the slope of the tangent line at \( x = \frac{3\pi}{4} \)
Substitute \( x = \frac{3\pi}{4} \) into the derivative \( f'(x) \):

\[
f'\left( \frac{3\pi}{4} \right) = -\cos\left( \frac{3\pi}{4} \right)
\]
Since \( \cos\left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2} \), we get:

\[
f'\left( \frac{3\pi}{4} \right) = -\left( -\frac{\sqrt{2}}{2} \right) = \frac{\sqrt{2}}{2}
\]

Thus, the slope of the tangent line is \( \frac{\sqrt{2}}{2} \).

### Step 3: Find the point \( \left( \frac{3\pi}{4}, f\left( \frac{3\pi}{4} \right) \right) \)
We now calculate the value of \( f(x) \) at \( x = \frac{3\pi}{4} \):

\[
f\left( \frac{3\pi}{4} \right) = -\sin\left( \frac{3\pi}{4} \right) = -\left( \frac{\sqrt{2}}{2} \right) = -\frac{\sqrt{2}}{2}
\]

Thus, the point on the curve is \( \left( \frac{3\pi}{4}, -\frac{\sqrt{2}}{2} \right) \).

### Step 4: Write the equation of the tangent line
Using the point-slope form of a line equation:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency, we substitute \( m = \frac{\sqrt{2}}{2} \), \( x_1 = \frac{3\pi}{4} \), and \( y_1 = -\frac{\sqrt{2}}{2} \):

\[
y - \left( -\frac{\sqrt{2}}{2} \right) = \frac{\sqrt{2}}{2} \left( x - \frac{3\pi}{4} \right)
\]

Simplifying:
\[
y + \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \left( x - \frac{3\pi}{4} \right)
\]

This is the equation of the tangent line.

Would you like a detailed explanation of any part of this solution?

### Related Questions:
1. How do you find the equation of a tangent line to a different trigonometric function, like \( \cos(x) \)?
2. What is the geometric interpretation of the derivative at a point on a curve?
3. How does the slope of a tangent line relate to increasing or decreasing functions?
4. Can a tangent line be horizontal? Under what conditions does this happen?
5. How would the tangent line equation change if the point was \( x = \pi/2 \)?

**Tip:** When calculating trigonometric values, it's useful to know the exact values for common angles like \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), etc.

Given f(x) = -sin(x), write the equation of the line tangent to y = f(x) when x = 3π/4.

Find the equation of the tangent line to the function f(x) = -sin(x) at x = 3π/4 using derivatives and the point-slope form. This solution involves calculus and trigonometric concepts, suitable for high school students studying Calculus.

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