To find the equation of the tangent line to the parametric curve defined by $x = 5\sin(t)$ and $y = 5\cos(t)$ at the point where $t = \frac{3\pi}{4}$, follow these steps:

1. Find the coordinates of the point on the curve:

Substitute $t = \frac{3\pi}{4}$ into the parametric equations for $x$ and $y$:

$x = 5\sin\left(\frac{3\pi}{4}\right)$ $y = 5\cos\left(\frac{3\pi}{4}\right)$

Evaluate these:

$\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}$ $\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

So the coordinates of the point are:

$x = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2}$ $y = 5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{2}}{2}$

2. Find the slope of the tangent line:

The slope of the tangent line to the parametric curve is given by $\frac{dy}{dx}$. We find this by differentiating $y$ and $x$ with respect to $t$:

$\frac{dx}{dt} = \frac{d}{dt}(5\sin t) = 5\cos t$ $\frac{dy}{dt} = \frac{d}{dt}(5\cos t) = -5\sin t$

So the slope $\frac{dy}{dx}$ is:

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{-5\sin t}{5\cos t} = -\tan t$

At $t = \frac{3\pi}{4}$:

$\tan\left(\frac{3\pi}{4}\right) = -1$

Thus, the slope of the tangent line is $1$.

3. Write the equation of the tangent line:

The equation of the tangent line can be written in point-slope form:

$y - y_1 = m(x - x_1)$

Substituting the point $\left(\frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2}\right)$ and the slope $m = 1$:

$y + \frac{5\sqrt{2}}{2} = 1\left(x - \frac{5\sqrt{2}}{2}\right)$

Simplifying, we get:

$y = x - 5\sqrt{2}$

4. Compare with the given options:

The correct answer is:

D. $y = x - \frac{5\sqrt{2}}{2}$

Would you like further details or have any questions?

Here are some additional related questions:

1. What is the general process for finding the slope of a tangent line in parametric form?
2. How does the point-slope form of a line equation work?
3. Why is $\tan\left(\frac{3\pi}{4}\right)$ equal to $-1$?
4. How can you verify the slope of a curve at a specific point using derivatives?
5. What is the significance of the parametric form in describing curves?

Tip: Always ensure the angle is correctly evaluated when dealing with trigonometric functions, as they significantly affect the slope and the final equation of the tangent line.