To solve the given problem, we need to find $\frac{dy}{dx}$ when $t = \frac{\pi}{4}$.

Given the parametric equations: $x = \sin t$ $y = t \sin t + \cos t$

We will follow these steps:

1. Differentiate $x$ and $y$ with respect to $t$.
2. Use the chain rule to find $\frac{dy}{dx}$.

First, let's find $\frac{dx}{dt}$: $\frac{dx}{dt} = \cos t$

Next, let's find $\frac{dy}{dt}$: $\frac{dy}{dt} = \frac{d}{dt} (t \sin t + \cos t)$ Using the product rule and chain rule: $\frac{dy}{dt} = \sin t + t \cos t - \sin t$ $\frac{dy}{dt} = t \cos t$

Now, using the chain rule: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{t \cos t}{\cos t} = t$

Finally, we substitute $t = \frac{\pi}{4}$ into the result: $\frac{dy}{dx} \bigg|_{t = \frac{\pi}{4}} = \frac{\pi}{4}$

So, the value of $\frac{dy}{dx}$ when $t = \frac{\pi}{4}$ is $\frac{\pi}{4}$.

Would you like any further details or have any questions? Here are some related questions you might want to explore:

1. How do you find the second derivative $\frac{d^2y}{dx^2}$ for parametric equations?
2. What are some applications of parametric equations in real-world problems?
3. Can you provide examples of parametric equations in polar coordinates?
4. How do you convert parametric equations to Cartesian form?
5. What are some common techniques for solving differential equations?

Tip: When working with parametric equations, always remember to use the chain rule for differentiation to relate the derivatives with respect to different variables.